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It shares its Lie algebra with SU(3), with a different global form.", "definition": "SU(3) modulo scalar cube roots of unity", "definitionEvidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "8", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: dim su(3)=8; a discrete central quotient preserves the Lie algebra" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "Trivial", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.1: the compact adjoint form has trivial center" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group", "value": "ℤ₃", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "rank", "label": "Rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representations", "paragraphs": [ { "text": "An SU(3) irrep (p,q) descends precisely when p+2q is zero modulo 3. 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The scalar center generator exp(2πi/3)I acts with charge p + 2q modulo 3.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2: adjoint-group weights are the root lattice" }, { "source": "etingof-lie-notes", "locator": "Section 27.2: the two A₂ fundamental modules have scalar charges 1 and 2" } ], "derivation": { "method": "The defining module has charge 1 and its exterior square has charge 2; highest-weight charges add.", "inputs": [ "Fundamental modules C³ and Λ²C³", "Dynkin labels p,q" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "related", "target": "group:su-3", "label": "Related group" }, { "type": "has-algebra", "target": "algebra:su-3-r", "label": "Actual real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:su-3-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:su-3-to-psu-3", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ] } ], "capabilities": [ "global-forms" ], "kind": "group", "data": { "category": "lie", "dimension": 8, "compact": true, "connected": true, "algebraId": "algebra:su-3-r" } }, { "id": "group:u-1", "slug": "u-1", "name": "Circle group", "symbol": "U(1)", "category": "lie", "level": "reference", "aliases": [ "Circle", "SO(2)" ], "searchTerms": [], "tags": [ "Charges", "Gauge theory" ], "summary": "The compact connected abelian Lie group of phases. It is the basic model for charge quantization after choosing a period.", "definition": "Complex numbers z with |z| = 1", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 6: U(1) has Lie algebra iR of real dimension 1" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "U(1)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group", "value": "ℤ", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: R → R/Z as a covering of the circle" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "rank", "label": "Rank", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representations", "paragraphs": [ { "text": "Continuous irreducible complex representations are eⁱθ ↦ eⁱⁿθ with n ∈ ℤ, for θ of period 2π.", "evidence": [ { "source": "conrad", "locator": "Lecture 27: characters of a torus; U(1) character lattice Z" } ], "assumptions": [] } ] }, { "id": "topology", "title": "Topology and global form", "paragraphs": [ { "text": "U(1) is a circle S¹. Its universal covering group is the additive group ℝ.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering groups of the circle" } ], "assumptions": [] } ] }, { "id": "physics", "title": "In physics", "paragraphs": [ { "text": "Electromagnetism, phase symmetries, maximal tori, and charge conventions.", "evidence": [ { "source": "conrad", "locator": "Lecture 27: characters of a torus; U(1) character lattice Z" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "related", "target": "group:su-2", "label": "Related group" }, { "type": "related", "target": "group:so-3", "label": "Related group" }, { "type": "has-algebra", "target": "algebra:u-1-r", "label": "Actual real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:u-1-to-su-2", "label": "Explicit homomorphism", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "has-map", "target": "map:u-1-to-so-3", "label": "Explicit homomorphism", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 1, "compact": true, "connected": true, "algebraId": "algebra:u-1-r" } }, { "id": "group:spin-4", "slug": "spin-4", "name": "Euclidean spin group in four dimensions", "symbol": "Spin(4)", "category": "lie", "level": "reference", "aliases": [ "SU(2) × SU(2)" ], "searchTerms": [], "tags": [ "Spinors", "Euclidean fields" ], "summary": "The simply connected double cover of SO(4), isomorphic to SU(2) × SU(2). 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Descent to SO(4) requires jL+jR ∈ ℤ.", "evidence": [ { "source": "conrad", "locator": "Lecture 27: representations of a product group" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and the diagonal order-two SO(4) kernel" } ], "assumptions": [] } ] }, { "id": "topology", "title": "Topology and global form", "paragraphs": [ { "text": "Spin(4) is diffeomorphic to S³ × S³. The kernel of the map to SO(4) is the diagonal ℤ₂.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic product model and diagonal kernel" } ], "assumptions": [] } ] }, { "id": "physics", "title": "In physics", "paragraphs": [ { "text": "Euclidean Weyl spinors, self-dual two-forms, and four-dimensional geometry.", "evidence": [ { "source": "haber", "locator": "Applications to the indicated symmetry and representation category" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "related", "target": "group:su-2", "label": "Related group" }, { "type": "related", "target": "group:sl-2-c", "label": "Related group" }, { "type": "has-algebra", "target": "algebra:so-4-compact-r", "label": "Actual real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:so-4-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:spin-4-to-so-4", "label": "Explicit homomorphism", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 6, "compact": true, "connected": true, "algebraId": "algebra:so-4-compact-r" } }, { "id": "group:sl-2-c", "slug": "sl-2-c", "name": "Lorentz spin group", "symbol": "SL(2,ℂ)", "category": "lie", "level": "reference", "aliases": [ "Spin+(3,1)", "Lorentz spin cover" ], "searchTerms": [], "tags": [ "Lorentz", "Spinors" ], "summary": "As a real Lie group, SL(2,ℂ) is the double cover of the identity component SO⁺(3,1) of the Lorentz group.", "definition": "2 × 2 complex matrices of determinant 1", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: sl(2,C) has complex dimension 3 and real dimension 6" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "ℤ₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: the SL(2,C) spin covering kernel {±I}" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group", "value": "0", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] }, { "key": "rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "knapp-structure", "locator": "Theorem 6.105: complex simple algebras as real algebras; the split diagonal subgroup of SL(2,C) has real rank 1" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representations", "paragraphs": [ { "text": "Finite-dimensional irreducible complex representations of the real Lie group carry labels (jL,jR). 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It is simply connected and noncompact.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ] }, { "id": "physics", "title": "In physics", "paragraphs": [ { "text": "Weyl and Dirac fields, Lorentz covariance, and the distinction between fields and states.", "evidence": [ { "source": "zuber", "locator": "Applications to the indicated symmetry and representation category" } ], "assumptions": [] } ] }, { "id": "real-algebra", "title": "The Lorentz algebra is a realification", "paragraphs": [ { "text": "SL(2,C) is being used as a real six-dimensional Lie group. Its real Lie algebra complexifies to sl(2,C) ⊕ sl(2,C), rather than to a single A₁ factor. 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the untwisted finite group of Lie type Aₙ₋₁.", "definitionEvidence": [ { "source": "finite-wilson-linear", "locator": "Definitions; more examples; simplicity of PSLₙ(q)" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 2; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-linear", "locator": "Definitions; more examples; simplicity of PSLₙ(q)" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Exclude (n,q) = (2,2) and (2,3).", "evidence": [ { "source": "finite-wilson-linear", "locator": "Definitions; more examples; simplicity of PSLₙ(q)" } ], "assumptions": [] }, { "key": "order-formula", "label": "Order", "value": "q^[n(n−1)/2] · ∏ᵢ₌₂ⁿ(qⁱ−1) / gcd(n,q−1)", "evidence": [ { "source": "finite-wilson-linear", "locator": "Orders of linear groups and scalar quotient" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "(2,2), (2,3): PSL(2,2) ≅ S₃ and PSL(2,3) ≅ A₄ are not simple.", "evidence": [ { "source": "finite-wilson-linear", "locator": "Definitions; more examples; simplicity of PSLₙ(q)" } ], "assumptions": [] }, { "text": "(2,4), (2,5): Both identify with A₅.", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [] }, { "text": "(2,7), (3,2): PSL(2,7) ≅ PSL(3,2).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [] }, { "text": "(2,9): PSL(2,9) ≅ A₆.", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [] }, { "text": "(4,2): PSL(4,2) ≅ A₈.", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "related-to", "target": "group:a-5", "label": "Canonical A₅ record", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ] }, { "type": "related-to", "target": "group:psl-2-7", "label": "Canonical PSL(2,7) record", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ] }, { "type": "related-to", "target": "group:a-6", "label": "Canonical A₆ record", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "integer n ≥ 2" }, { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [ { "at": "(2,2), (2,3)", "description": "PSL(2,2) ≅ S₃ and PSL(2,3) ≅ A₄ are not simple." }, { "at": "(2,4), (2,5)", "description": "Both identify with A₅." }, { "at": "(2,7), (3,2)", "description": "PSL(2,7) ≅ PSL(3,2)." }, { "at": "(2,9)", "description": "PSL(2,9) ≅ A₆." }, { "at": "(4,2)", "description": "PSL(4,2) ≅ A₈." } ], "scope": "Exclude (n,q) = (2,2) and (2,3)." } }, { "id": "family:finite-unitary", "slug": "finite-unitary", "name": "Projective special unitary finite simple groups", "symbol": "PSU(n,q)", "category": "finite", "level": "classification", "aliases": [ "Uₙ(q)", "²Aₙ₋₁(q)" ], "searchTerms": [ "Projective special unitary finite simple groups", "PSU(n,q)", "Uₙ(q)", "²Aₙ₋₁(q)" ], "tags": [ "Finite simple classification" ], "summary": "Exclude (n,q) = (3,2); q is the fixed-field size, not the matrix-field size.", "definition": "The determinant-one isometries of a nondegenerate Hermitian form over F(q²), modulo scalar matrices.", "definitionEvidence": [ { "source": "finite-wilson-classical", "locator": "Unitary groups; structure of unitary groups I–II" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 3; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-classical", "locator": "Unitary groups; structure of unitary groups I–II" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Exclude (n,q) = (3,2); q is the fixed-field size, not the matrix-field size.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Unitary groups; structure of unitary groups I–II" } ], "assumptions": [] }, { "key": "order-formula", "label": "Order", "value": "q^[n(n−1)/2] · ∏ᵢ₌₂ⁿ(qⁱ−(−1)ⁱ) / gcd(n,q+1)", "evidence": [ { "source": "finite-wilson-classical", "locator": "Orders and scalar center of unitary groups" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "n = 2: PSU(2,q) ≅ PSL(2,q); 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q is the fixed-field size, not the matrix-field size." } }, { "id": "family:finite-symplectic", "slug": "finite-symplectic", "name": "Projective symplectic finite simple groups", "symbol": "PSp(2n,q)", "category": "finite", "level": "classification", "aliases": [ "Cₙ(q)", "S₂ₙ(q)" ], "searchTerms": [ "Projective symplectic finite simple groups", "PSp(2n,q)", "Cₙ(q)", "S₂ₙ(q)" ], "tags": [ "Finite simple classification" ], "summary": "Exclude (n,q) = (2,2).", "definition": "The isometries of a nondegenerate alternating form on Fq^(2n), modulo scalar matrices.", "definitionEvidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.2.1–3.2.2" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 2; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.2.1–3.2.2" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Exclude (n,q) = (2,2).", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.2.1–3.2.2" } ], "assumptions": [] }, { "key": "order-formula", "label": "Order", "value": "q^(n²) · ∏ᵢ₌₁ⁿ(q^(2i)−1) / gcd(2,q−1)", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.2.1, order and scalar quotient" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "n = 1: PSp(2,q) ≅ PSL(2,q); use the linear family.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.2.1–3.2.2" } ], "assumptions": [] }, { "text": "(2,2): Sp(4,2) ≅ S₆; its derived subgroup is A₆.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.2.2" }, { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [] }, { "text": "(2,3): PSp(4,3) ≅ PSU(4,2).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "related-to", "target": "group:a-6", "label": "A₆ as the derived small case", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "integer n ≥ 2" }, { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [ { "at": "n = 1", "description": "PSp(2,q) ≅ PSL(2,q); use the linear family." }, { "at": "(2,2)", "description": "Sp(4,2) ≅ S₆; its derived subgroup is A₆." }, { "at": "(2,3)", "description": "PSp(4,3) ≅ PSU(4,2)." } ], "scope": "Exclude (n,q) = (2,2)." } }, { "id": "family:finite-odd-orthogonal", "slug": "finite-odd-orthogonal", "name": "Odd-dimensional orthogonal finite simple groups", "symbol": "PΩ(2n+1,q)", "category": "finite", "level": "classification", "aliases": [ "Bₙ(q)", "Ω₂ₙ₊₁(q)" ], "searchTerms": [ "Odd-dimensional orthogonal finite simple groups", "PΩ(2n+1,q)", "Bₙ(q)", "Ω₂ₙ₊₁(q)" ], "tags": [ "Finite simple classification" ], "summary": "Use odd q and n ≥ 3 to avoid the symplectic and low-rank repetitions.", "definition": "The projective Ω group of a nondegenerate quadratic form in odd dimension over Fq.", "definitionEvidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.4, orthogonal groups in odd characteristic" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 3; q: odd prime power", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.4, orthogonal groups in odd characteristic" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Use odd q and n ≥ 3 to avoid the symplectic and low-rank repetitions.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.4, orthogonal groups in odd characteristic" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "dimension 3: PΩ(3,q) ≅ PSL(2,q).", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup: B-type conventions" } ], "assumptions": [] }, { "text": "dimension 5: PΩ(5,q) ≅ PSp(4,q).", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [] }, { "text": "even q: The odd-dimensional orthogonal groups identify with the corresponding symplectic groups.", "evidence": [ { "source": "finite-gap-classical-maximals", "locator": "Section 2.2-4, first paragraph" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "Ω denotes the subgroup required for simplicity; merely taking a projective special orthogonal group does not in general give the simple group.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.4, orthogonal groups in odd characteristic" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "integer n ≥ 3" }, { "name": "q", "domain": "odd prime power" } ], "exceptions": [ { "at": "dimension 3", "description": "PΩ(3,q) ≅ PSL(2,q)." }, { "at": "dimension 5", "description": "PΩ(5,q) ≅ PSp(4,q)." }, { "at": "even q", "description": "The odd-dimensional orthogonal groups identify with the corresponding symplectic groups." } ], "scope": "Use odd q and n ≥ 3 to avoid the symplectic and low-rank repetitions." } }, { "id": "family:finite-even-orthogonal-plus", "slug": "finite-even-orthogonal-plus", "name": "Plus-type even orthogonal finite simple groups", "symbol": "PΩ⁺(2n,q)", "category": "finite", "level": "classification", "aliases": [ "Dₙ(q)", "O₂ₙ⁺(q)" ], "searchTerms": [ "Plus-type even orthogonal finite simple groups", "PΩ⁺(2n,q)", "Dₙ(q)", "O₂ₙ⁺(q)" ], "tags": [ "Finite simple classification" ], "summary": "n ≥ 4; plus type has Witt index n.", "definition": "The projective Ω group of a nondegenerate split quadratic form on Fq^(2n).", "definitionEvidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, plus type and projective Ω groups" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 4; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, plus type and projective Ω groups" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "n ≥ 4; plus type has Witt index n.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, plus type and projective Ω groups" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "dimension 4: PΩ⁺(4,q) ≅ PSL(2,q) × PSL(2,q), so this is not a new simple family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [] }, { "text": "dimension 6: PΩ⁺(6,q) ≅ PSL(4,q); use the linear family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "integer n ≥ 4" }, { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [ { "at": "dimension 4", "description": "PΩ⁺(4,q) ≅ PSL(2,q) × PSL(2,q), so this is not a new simple family." }, { "at": "dimension 6", "description": "PΩ⁺(6,q) ≅ PSL(4,q); use the linear family." } ], "scope": "n ≥ 4; plus type has Witt index n." } }, { "id": "family:finite-even-orthogonal-minus", "slug": "finite-even-orthogonal-minus", "name": "Minus-type even orthogonal finite simple groups", "symbol": "PΩ⁻(2n,q)", "category": "finite", "level": "classification", "aliases": [ "²Dₙ(q)", "O₂ₙ⁻(q)" ], "searchTerms": [ "Minus-type even orthogonal finite simple groups", "PΩ⁻(2n,q)", "²Dₙ(q)", "O₂ₙ⁻(q)" ], "tags": [ "Finite simple classification" ], "summary": "n ≥ 4; minus type has Witt index n−1.", "definition": "The projective Ω group of a nonsplit nondegenerate quadratic form on Fq^(2n).", "definitionEvidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, minus type and projective Ω groups" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 4; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, minus type and projective Ω groups" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "n ≥ 4; minus type has Witt index n−1.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, minus type and projective Ω groups" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "dimension 4: PΩ⁻(4,q) ≅ PSL(2,q²); use the linear family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [] }, { "text": "dimension 6: PΩ⁻(6,q) ≅ PSU(4,q); use the unitary family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "integer n ≥ 4" }, { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [ { "at": "dimension 4", "description": "PΩ⁻(4,q) ≅ PSL(2,q²); use the linear family." }, { "at": "dimension 6", "description": "PΩ⁻(6,q) ≅ PSU(4,q); use the unitary family." } ], "scope": "n ≥ 4; minus type has Witt index n−1." } }, { "id": "family:finite-g2", "slug": "finite-g2", "name": "Finite simple groups of type G₂", "symbol": "G₂(q)", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "Finite simple groups of type G₂", "G₂(q)" ], "tags": [ "Finite simple classification" ], "summary": "q ≥ 3; the q = 2 parent group is not simple.", "definition": "The untwisted Chevalley group of type G₂ over Fq.", "definitionEvidence": [ { "source": "finite-wilson-exceptional", "locator": "G₂(q), order and simplicity" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: prime power q ≥ 3", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "G₂(q), order and simplicity" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "q ≥ 3; the q = 2 parent group is not simple.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "G₂(q), order and simplicity" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "q = 2: G₂(2)′ ≅ PSU(3,3), of index 2 in G₂(2).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" }, { "source": "finite-wilson-exceptional", "locator": "G₂(q), small case" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "prime power q ≥ 3" } ], "exceptions": [ { "at": "q = 2", "description": "G₂(2)′ ≅ PSU(3,3), of index 2 in G₂(2)." } ], "scope": "q ≥ 3; the q = 2 parent group is not simple." } }, { "id": "family:finite-f4", "slug": "finite-f4", "name": "Finite simple groups of type F₄", "symbol": "F₄(q)", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "Finite simple groups of type F₄", "F₄(q)" ], "tags": [ "Finite simple classification" ], "summary": "All prime powers q ≥ 2; take the simple central quotient.", "definition": "The untwisted Chevalley finite group of Lie type F₄, in its simple central-quotient form.", "definitionEvidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [], "scope": "All prime powers q ≥ 2; take the simple central quotient." } }, { "id": "family:finite-e6", "slug": "finite-e6", "name": "Finite simple groups of type E₆", "symbol": "E₆(q)", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "Finite simple groups of type E₆", "E₆(q)" ], "tags": [ "Finite simple classification" ], "summary": "All prime powers q ≥ 2; take the simple central quotient.", "definition": "The untwisted Chevalley finite group of Lie type E₆, in its simple central-quotient form.", "definitionEvidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [], "scope": "All prime powers q ≥ 2; take the simple central quotient." } }, { "id": "family:finite-e7", "slug": "finite-e7", "name": "Finite simple groups of type E₇", "symbol": "E₇(q)", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "Finite simple groups of type E₇", "E₇(q)" ], "tags": [ "Finite simple classification" ], "summary": "All prime powers q ≥ 2; take the simple central quotient.", "definition": "The untwisted Chevalley finite group of Lie type E₇, in its simple central-quotient form.", "definitionEvidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [], "scope": "All prime powers q ≥ 2; take the simple central quotient." } }, { "id": "family:finite-e8", "slug": "finite-e8", "name": "Finite simple groups of type E₈", "symbol": "E₈(q)", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "Finite simple groups of type E₈", "E₈(q)" ], "tags": [ "Finite simple classification" ], "summary": "All prime powers q ≥ 2; take the simple central quotient.", "definition": "The untwisted Chevalley finite group of Lie type E₈, in its simple central-quotient form.", "definitionEvidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [], "scope": "All prime powers q ≥ 2; take the simple central quotient." } }, { "id": "family:finite-twisted-e6", "slug": "finite-twisted-e6", "name": "Finite simple groups of type ²E₆", "symbol": "²E₆(q)", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "Finite simple groups of type ²E₆", "²E₆(q)" ], "tags": [ "Finite simple classification" ], "summary": "All prime powers q ≥ 2; take the simple central quotient.", "definition": "The graph-field twisted Steinberg finite group of Lie type ²E₆, in its simple central-quotient form.", "definitionEvidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [], "scope": "All prime powers q ≥ 2; take the simple central quotient." } }, { "id": "family:finite-triality", "slug": "finite-triality", "name": "Finite simple groups of type ³D₄", "symbol": "³D₄(q)", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "Finite simple groups of type ³D₄", "³D₄(q)" ], "tags": [ "Finite simple classification" ], "summary": "All prime powers q ≥ 2; take the simple central quotient.", "definition": "The triality-twisted Steinberg finite group of Lie type ³D₄, in its simple central-quotient form.", "definitionEvidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = p^f, with p prime and f ≥ 1" } ], "exceptions": [], "scope": "All prime powers q ≥ 2; take the simple central quotient." } }, { "id": "family:finite-suzuki", "slug": "finite-suzuki", "name": "Suzuki simple groups", "symbol": "²B₂(q)", "category": "finite", "level": "classification", "aliases": [ "Sz(q)" ], "searchTerms": [ "Suzuki simple groups", "²B₂(q)", "Sz(q)" ], "tags": [ "Finite simple classification" ], "summary": "Odd extension degree in characteristic 2; q ≥ 8.", "definition": "The Suzuki twisted groups of type ²B₂.", "definitionEvidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Suzuki construction and simplicity; section 5" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = 2^(2m+1), integer m ≥ 1", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Suzuki construction and simplicity; section 5" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Odd extension degree in characteristic 2; q ≥ 8.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Suzuki construction and simplicity; section 5" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "m = 0; q = 2: Sz(2) has order 20 and is not simple.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree groups: small cases" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "Sz(q) denotes this infinite family; Suz denotes the distinct sporadic Suzuki group.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Suzuki construction and simplicity; section 5" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "related-to", "target": "group:sporadic-suz", "label": "Sporadic Suz: different group", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Suz entry" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = 2^(2m+1), integer m ≥ 1" } ], "exceptions": [ { "at": "m = 0; q = 2", "description": "Sz(2) has order 20 and is not simple." } ], "scope": "Odd extension degree in characteristic 2; q ≥ 8." } }, { "id": "family:finite-ree-g2", "slug": "finite-ree-g2", "name": "Small Ree simple groups", "symbol": "²G₂(q)", "category": "finite", "level": "classification", "aliases": [ "Ree(q)" ], "searchTerms": [ "Small Ree simple groups", "²G₂(q)", "Ree(q)" ], "tags": [ "Finite simple classification" ], "summary": "Odd extension degree in characteristic 3; q ≥ 27.", "definition": "The Ree twisted groups of type ²G₂.", "definitionEvidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Small Ree construction and simplicity; section 5" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = 3^(2m+1), integer m ≥ 1", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Small Ree construction and simplicity; section 5" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Odd extension degree in characteristic 3; q ≥ 27.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Small Ree construction and simplicity; section 5" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "m = 0; q = 3: ²G₂(3)′ ≅ PSL(2,8), with index 3 in ²G₂(3).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" }, { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree groups: small cases" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = 3^(2m+1), integer m ≥ 1" } ], "exceptions": [ { "at": "m = 0; q = 3", "description": "²G₂(3)′ ≅ PSL(2,8), with index 3 in ²G₂(3)." } ], "scope": "Odd extension degree in characteristic 3; q ≥ 27." } }, { "id": "family:finite-ree-f4", "slug": "finite-ree-f4", "name": "Large Ree simple groups", "symbol": "²F₄(q)", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "Large Ree simple groups", "²F₄(q)" ], "tags": [ "Finite simple classification" ], "summary": "Odd extension degree in characteristic 2; q ≥ 8. The Tits group is a separate small-case entry.", "definition": "The Ree twisted groups of type ²F₄.", "definitionEvidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Large Ree construction and simplicity; section 5" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "q: q = 2^(2m+1), integer m ≥ 1", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Large Ree construction and simplicity; section 5" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Odd extension degree in characteristic 2; q ≥ 8. The Tits group is a separate small-case entry.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Large Ree construction and simplicity; section 5" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "m = 0; q = 2: ²F₄(2) is not simple. Its index-two derived subgroup ²F₄(2)′ is the Tits simple group.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree groups: small cases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "related-to", "target": "group:tits", "label": "Tits derived subgroup", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree groups: small cases" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "q", "domain": "q = 2^(2m+1), integer m ≥ 1" } ], "exceptions": [ { "at": "m = 0; q = 2", "description": "²F₄(2) is not simple. Its index-two derived subgroup ²F₄(2)′ is the Tits simple group." } ], "scope": "Odd extension degree in characteristic 2; q ≥ 8. The Tits group is a separate small-case entry." } }, { "id": "family:finite-sporadic", "slug": "finite-sporadic", "name": "The 26 sporadic finite simple groups", "symbol": "Sporadic", "category": "finite", "level": "classification", "aliases": [], "searchTerms": [ "The 26 sporadic finite simple groups", "Sporadic" ], "tags": [ "Finite simple classification" ], "summary": "Exactly 26 exceptional finite simple groups; the Tits group is excluded from this count.", "definition": "The finite list of nonabelian simple groups outside the alternating and Lie-type families.", "definitionEvidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "Finite collection", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Exactly 26 exceptional finite simple groups; the Tits group is excluded from this count.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement" } ], "assumptions": [] } ], "sections": [ { "id": "exceptions", "title": "Exceptions and identifications", "paragraphs": [ { "text": "Tits convention: ²F₄(2)′ is indexed with the Lie-type small cases, not as a 27th sporadic group.", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Miscellaneous: Tits group" }, { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree small cases" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "Individual entries record identity and order. The linked ATLAS separates representations by group, field and characteristic.", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Sporadic index and individual group pages" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [], "exceptions": [ { "at": "Tits convention", "description": "²F₄(2)′ is indexed with the Lie-type small cases, not as a 27th sporadic group." } ], "scope": "Exactly 26 exceptional finite simple groups; the Tits group is excluded from this count." } }, { "id": "family:finite-cyclic", "slug": "finite-cyclic", "name": "Finite cyclic groups", "symbol": "Cₙ", "category": "finite", "level": "reference", "aliases": [ "Z/nZ", "ℤₙ" ], "searchTerms": [ "Finite cyclic groups", "Cₙ", "Z/nZ", "ℤₙ" ], "tags": [ "Finite families" ], "summary": "One cyclic group for every finite order n.", "definition": "The group generated by a with relation aⁿ = 1, with a of exact order n.", "definitionEvidence": [ { "source": "finite-gap-constructors", "locator": "CyclicGroup" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: positive integer", "evidence": [ { "source": "finite-gap-constructors", "locator": "CyclicGroup" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "One cyclic group for every finite order n.", "evidence": [ { "source": "finite-gap-constructors", "locator": "CyclicGroup" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "Ordinary complex irreducibles are the one-dimensional characters χₖ(a) = exp(2πik/n), with k taken modulo n.", "evidence": [ { "source": "gap", "locator": "Ordinary character tables and irreducible characters" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "positive integer" } ], "exceptions": [], "scope": "One cyclic group for every finite order n." } }, { "id": "family:finite-dihedral", "slug": "finite-dihedral", "name": "Finite dihedral groups", "symbol": "D₂ₙ", "category": "finite", "level": "reference", "aliases": [], "searchTerms": [ "Finite dihedral groups", "D₂ₙ" ], "tags": [ "Finite families" ], "summary": "This site writes D₂ₙ for the group of order 2n.", "definition": "Symmetries of a regular n-gon: rⁿ = s² = 1 and srs⁻¹ = r⁻¹.", "definitionEvidence": [ { "source": "finite-gap-constructors", "locator": "DihedralGroup" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 3", "evidence": [ { "source": "finite-gap-constructors", "locator": "DihedralGroup" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "This site writes D₂ₙ for the group of order 2n.", "evidence": [ { "source": "finite-gap-constructors", "locator": "DihedralGroup" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "A competing convention calls the same group Dₙ. Check the order whenever importing a dihedral label.", "evidence": [ { "source": "finite-gap-constructors", "locator": "DihedralGroup" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "integer n ≥ 3" } ], "exceptions": [], "scope": "This site writes D₂ₙ for the group of order 2n." } }, { "id": "family:finite-dicyclic", "slug": "finite-dicyclic", "name": "Dicyclic groups", "symbol": "Dicₙ", "category": "finite", "level": "reference", "aliases": [ "Binary dihedral groups" ], "searchTerms": [ "Dicyclic groups", "Dicₙ", "Binary dihedral groups" ], "tags": [ "Finite families" ], "summary": "Dicₙ has order 4n; Dic₂ = Q₈ and Dic₄ = Q₁₆.", "definition": "The group with a^(2n) = 1, x² = aⁿ and xax⁻¹ = a⁻¹.", "definitionEvidence": [ { "source": "finite-gap-constructors", "locator": "Sections 50.1-8 DicyclicGroup and 50.1-9 IsDicyclicGroup" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 2", "evidence": [ { "source": "finite-gap-constructors", "locator": "Sections 50.1-8 DicyclicGroup and 50.1-9 IsDicyclicGroup" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "Dicₙ has order 4n; Dic₂ = Q₈ and Dic₄ = Q₁₆.", "evidence": [ { "source": "finite-gap-constructors", "locator": "Sections 50.1-8 DicyclicGroup and 50.1-9 IsDicyclicGroup" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "For n a power of two this is a generalized quaternion group.", "evidence": [ { "source": "finite-gap-constructors", "locator": "Sections 50.1-8 DicyclicGroup and 50.1-9 IsDicyclicGroup" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "integer n ≥ 2" } ], "exceptions": [], "scope": "Dicₙ has order 4n; Dic₂ = Q₈ and Dic₄ = Q₁₆." } }, { "id": "family:binary-polyhedral", "slug": "binary-polyhedral", "name": "Exceptional binary polyhedral groups", "symbol": "2T · 2O · 2I", "category": "finite", "level": "reference", "aliases": [], "searchTerms": [ "Exceptional binary polyhedral groups", "2T · 2O · 2I" ], "tags": [ "Finite families" ], "summary": "The binary tetrahedral, octahedral and icosahedral groups.", "definition": "The inverse images in SU(2) of the tetrahedral, octahedral and icosahedral rotation groups under SU(2) → SO(3).", "definitionEvidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups, slides 7–8" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "G: 2T, 2O or 2I", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups, slides 7–8" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "The binary tetrahedral, octahedral and icosahedral groups.", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups, slides 7–8" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" } ], "assumptions": [] } ], "sections": [], "relationships": [ { "type": "related-to", "target": "group:su-2", "label": "Finite subgroups of SU(2)", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups, slides 7–8" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "G", "domain": "2T, 2O or 2I" } ], "exceptions": [], "scope": "The binary tetrahedral, octahedral and icosahedral groups." } }, { "id": "family:finite-symmetric", "slug": "finite-symmetric", "name": "Finite symmetric groups", "symbol": "Sₙ", "category": "finite", "level": "reference", "aliases": [ "Sym(n)" ], "searchTerms": [ "Finite symmetric groups", "Sₙ", "Sym(n)" ], "tags": [ "Finite families" ], "summary": "All permutations of n symbols; order n!.", "definition": "The group of bijections of an n-element set, with composition as the operation.", "definitionEvidence": [ { "source": "finite-wilson-introduction", "locator": "Permutations; even permutations" } ], "properties": [ { "key": "parameters", "label": "Parameter domain", "value": "n: positive integer", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Permutations; even permutations" } ], "assumptions": [] }, { "key": "restriction", "label": "Scope and restrictions", "value": "All permutations of n symbols; order n!.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Permutations; even permutations" } ], "assumptions": [] } ], "sections": [ { "id": "conventions", "title": "Conventions and use", "paragraphs": [ { "text": "The sign homomorphism has kernel Aₙ; for n ≥ 5 that kernel is nonabelian simple.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Even permutations; simplicity of alternating groups" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "n", "domain": "positive integer" } ], "exceptions": [], "scope": "All permutations of n symbols; order n!." } }, { "id": "group:c-2", "slug": "c-2", "name": "Cyclic group of order 2", "symbol": "C2", "category": "finite", "level": "reference", "aliases": [ "Z2", "Z/2Z", "ℤ2" ], "searchTerms": [ "Cyclic group of order 2", "C2", "Z2", "Z/2Z", "ℤ2", "SmallGroup(2,1)" ], "tags": [ "Finite reference", "Finite simple", "Abelian" ], "summary": "A cyclic group with 2 one-dimensional ordinary complex irreducibles.", "definition": "The cyclic group generated by a with a^2 = 1 and exact order 2.", "definitionEvidence": [ { "source": "finite-groupnames-c-2", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "2", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C2", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Abelian label; upper central series" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Abelian label" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Simple label" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(2,1)", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "For a generator a, χₖ(a) = exp(2πik/2); source row labels need not be ordered by k.", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Character table" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-cyclic", "label": "Family", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Heading; Aliases" } ] }, { "type": "member-of", "target": "family:finite-prime-cyclic", "label": "Family", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Heading; Aliases" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "2", "abelian": true, "simple": true, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2" ], "dimensions": [ 1, 1 ], "reality": [ "Real", "Real" ] } } }, { "id": "group:c-4", "slug": "c-4", "name": "Cyclic group of order 4", "symbol": "C4", "category": "finite", "level": "reference", "aliases": [ "Z4", "Z/4Z", "ℤ4" ], "searchTerms": [ "Cyclic group of order 4", "C4", "Z4", "Z/4Z", "ℤ4", "SmallGroup(4,1)" ], "tags": [ "Finite reference", "Abelian" ], "summary": "A cyclic group with 4 one-dimensional ordinary complex irreducibles.", "definition": "The cyclic group generated by a with a^4 = 1 and exact order 4.", "definitionEvidence": [ { "source": "finite-groupnames-c-4", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "4", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C4", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Abelian label; upper central series" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Abelian label" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(4,1)", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "For a generator a, χₖ(a) = exp(2πik/4); source row labels need not be ordered by k.", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Character table" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-cyclic", "label": "Family", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Heading; Aliases" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "4", "abelian": true, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4" ], "dimensions": [ 1, 1, 1, 1 ], "reality": [ "Real", "Real", "Complex", "Complex" ] } } }, { "id": "group:c-5", "slug": "c-5", "name": "Cyclic group of order 5", "symbol": "C5", "category": "finite", "level": "reference", "aliases": [ "Z5", "Z/5Z", "ℤ5" ], "searchTerms": [ "Cyclic group of order 5", "C5", "Z5", "Z/5Z", "ℤ5", "SmallGroup(5,1)" ], "tags": [ "Finite reference", "Finite simple", "Abelian" ], "summary": "A cyclic group with 5 one-dimensional ordinary complex irreducibles.", "definition": "The cyclic group generated by a with a^5 = 1 and exact order 5.", "definitionEvidence": [ { "source": "finite-groupnames-c-5", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "5", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C5", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Abelian label; upper central series" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Abelian label" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Simple label" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 1", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(5,1)", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "For a generator a, χₖ(a) = exp(2πik/5); source row labels need not be ordered by k.", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Character table" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-cyclic", "label": "Family", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Heading; Aliases" } ] }, { "type": "member-of", "target": "family:finite-prime-cyclic", "label": "Family", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Heading; Aliases" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "5", "abelian": true, "simple": true, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5" ], "dimensions": [ 1, 1, 1, 1, 1 ], "reality": [ "Real", "Complex", "Complex", "Complex", "Complex" ] } } }, { "id": "group:c-6", "slug": "c-6", "name": "Cyclic group of order 6", "symbol": "C6", "category": "finite", "level": "reference", "aliases": [ "Z6", "Z/6Z", "ℤ6" ], "searchTerms": [ "Cyclic group of order 6", "C6", "Z6", "Z/6Z", "ℤ6", "SmallGroup(6,2)" ], "tags": [ "Finite reference", "Abelian" ], "summary": "A cyclic group with 6 one-dimensional ordinary complex irreducibles.", "definition": "The cyclic group generated by a with a^6 = 1 and exact order 6.", "definitionEvidence": [ { "source": "finite-groupnames-c-6", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "6", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C6", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Abelian label; upper central series" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Abelian label" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 1, 1", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(6,2)", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "For a generator a, χₖ(a) = exp(2πik/6); source row labels need not be ordered by k.", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Character table" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-cyclic", "label": "Family", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Heading; Aliases" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "6", "abelian": true, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6" ], "dimensions": [ 1, 1, 1, 1, 1, 1 ], "reality": [ "Real", "Real", "Complex", "Complex", "Complex", "Complex" ] } } }, { "id": "group:v-4", "slug": "v-4", "name": "Klein four-group", "symbol": "V₄", "category": "finite", "level": "reference", "aliases": [ "C2 × C2", "C₂²", "Klein group", "Z2 × Z2" ], "searchTerms": [ "Klein four-group", "V₄", "C2 × C2", "C₂²", "Klein group", "Z2 × Z2", "SmallGroup(4,2)" ], "tags": [ "Finite reference", "Abelian" ], "summary": "The noncyclic group of order four, with four real one-dimensional irreducibles.", "definition": "C₂ × C₂, generated by two commuting involutions.", "definitionEvidence": [ { "source": "finite-groupnames-v-4", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "4", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "V₄", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Abelian label; upper central series" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Abelian label" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(4,2)", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "Every nonidentity element has order two.", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "normal-subgroup-of", "target": "group:a-4", "label": "Normal Klein subgroup of A₄", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Subgroup lattice and maximal subgroup list" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "4", "abelian": true, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4" ], "dimensions": [ 1, 1, 1, 1 ], "reality": [ "Real", "Real", "Real", "Real" ] } } }, { "id": "group:d-10", "slug": "d-10", "name": "Dihedral group of order ten", "symbol": "D₁₀", "category": "finite", "level": "reference", "aliases": [ "Dihedral(10)", "Dih(5)", "D5 (polygon convention)" ], "searchTerms": [ "Dihedral group of order ten", "D₁₀", "Dihedral(10)", "Dih(5)", "D5 (polygon convention)", "SmallGroup(10,1)" ], "tags": [ "Finite reference", "Nonabelian" ], "summary": "Pentagon rotations and reflections, with two singlets and two real doublets.", "definition": "Symmetries of the regular pentagon: r⁵ = s² = 1 and srs⁻¹ = r⁻¹.", "definitionEvidence": [ { "source": "finite-groupnames-d-10", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "10", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 2, 2", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(10,1)", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "D₁₀ here means order ten. The cited GroupNames page uses D₅, with the subscript counting polygon vertices.", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-dihedral", "label": "Family", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Heading; Aliases" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "10", "abelian": false, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4" ], "dimensions": [ 1, 1, 2, 2 ], "reality": [ "Real", "Real", "Real", "Real" ] } } }, { "id": "group:d-12", "slug": "d-12", "name": "Dihedral group of order twelve", "symbol": "D₁₂", "category": "finite", "level": "reference", "aliases": [ "Dihedral(12)", "Dih(6)", "D6 (polygon convention)", "C2 × S3" ], "searchTerms": [ "Dihedral group of order twelve", "D₁₂", "Dihedral(12)", "Dih(6)", "D6 (polygon convention)", "C2 × S3", "SmallGroup(12,4)" ], "tags": [ "Finite reference", "Nonabelian" ], "summary": "Hexagon symmetries; abstractly C₂ × S₃, with entirely real ordinary irreducibles.", "definition": "Symmetries of the regular hexagon: r⁶ = s² = 1 and srs⁻¹ = r⁻¹.", "definitionEvidence": [ { "source": "finite-groupnames-d-12", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "12", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 2, 2", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(12,4)", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "The central half-turn generates C₂. Quotienting by the center gives S₃. The cited page calls this group D₆.", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-dihedral", "label": "Family", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Heading; Aliases" } ] }, { "type": "central-quotient", "target": "group:s-3", "label": "D₁₂ / C₂ ≅ S₃", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Centre: G/Z" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "12", "abelian": false, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6" ], "dimensions": [ 1, 1, 1, 1, 2, 2 ], "reality": [ "Real", "Real", "Real", "Real", "Real", "Real" ] } } }, { "id": "group:dic-3", "slug": "dic-3", "name": "Dicyclic group of order twelve", "symbol": "Dic₃", "category": "finite", "level": "reference", "aliases": [ "Dic(3)", "C3 ⋊ C4", "Binary dihedral group of order 12" ], "searchTerms": [ "Dicyclic group of order twelve", "Dic₃", "Dic(3)", "C3 ⋊ C4", "Binary dihedral group of order 12", "SmallGroup(12,1)" ], "tags": [ "Finite reference", "Nonabelian" ], "summary": "A central double cover of S₃ with one real and one quaternionic doublet.", "definition": "The group with a⁶ = 1, x² = a³ and xax⁻¹ = a⁻¹.", "definitionEvidence": [ { "source": "finite-groupnames-dic-3", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "12", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 2, 2", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(12,1)", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "Its abelianization is C₄; two singlets form a complex-conjugate pair.", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-dicyclic", "label": "Family", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Heading; Aliases" } ] }, { "type": "central-quotient", "target": "group:s-3", "label": "Dic₃ / C₂ ≅ S₃", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Centre: G/Z" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "12", "abelian": false, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6" ], "dimensions": [ 1, 1, 1, 1, 2, 2 ], "reality": [ "Real", "Real", "Complex", "Complex", "Real", "Quaternionic" ] } } }, { "id": "group:q-16", "slug": "q-16", "name": "Generalized quaternion group of order sixteen", "symbol": "Q₁₆", "category": "finite", "level": "reference", "aliases": [ "QuaternionGroup(16)", "Dic4", "Dic(4)" ], "searchTerms": [ "Generalized quaternion group of order sixteen", "Q₁₆", "QuaternionGroup(16)", "Dic4", "Dic(4)", "SmallGroup(16,9)" ], "tags": [ "Finite reference", "Nonabelian" ], "summary": "A generalized quaternion group with four real singlets, one real doublet and two quaternionic doublets.", "definition": "The group with a⁸ = 1, x² = a⁴ and xax⁻¹ = a⁻¹.", "definitionEvidence": [ { "source": "finite-groupnames-q-16", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "16", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 2, 2, 2", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(16,9)", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-dicyclic", "label": "Family", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Heading; Aliases" } ] }, { "type": "central-quotient", "target": "group:d-8", "label": "Q₁₆ / C₂ ≅ D₈", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Centre: quotient D₄ in the order-eight convention" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "16", "abelian": false, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6", "ρ7" ], "dimensions": [ 1, 1, 1, 1, 2, 2, 2 ], "reality": [ "Real", "Real", "Real", "Real", "Real", "Quaternionic", "Quaternionic" ] } } }, { "id": "group:binary-tetrahedral", "slug": "binary-tetrahedral", "name": "Binary tetrahedral group", "symbol": "2T", "category": "finite", "level": "reference", "aliases": [ "SL(2,3)", "2.A4", "Q8 ⋊ C3" ], "searchTerms": [ "Binary tetrahedral group", "2T", "SL(2,3)", "2.A4", "Q8 ⋊ C3", "SmallGroup(24,3)" ], "tags": [ "Finite reference", "Nonabelian" ], "summary": "The central double cover of A₄, with three doublets of different Frobenius–Schur types.", "definition": "The inverse image of the tetrahedral rotation group in SU(2); abstractly SL(2,3).", "definitionEvidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups; binary tetrahedral example" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases" } ], "properties": [ { "key": "order", "label": "Order", "value": "24", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 2, 2, 2, 3", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(24,3)", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "The center acts trivially on the three singlets and the triplet; those representations descend to A₄.", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:binary-polyhedral", "label": "Family", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Heading; Aliases" } ] }, { "type": "central-quotient", "target": "group:a-4", "label": "2T / C₂ ≅ A₄", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Centre: G/Z" } ] }, { "type": "subgroup-of", "target": "group:su-2", "label": "Binary tetrahedral embedding", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "24", "abelian": false, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6", "ρ7" ], "dimensions": [ 1, 1, 1, 2, 2, 2, 3 ], "reality": [ "Real", "Complex", "Complex", "Quaternionic", "Complex", "Complex", "Real" ] } } }, { "id": "group:binary-octahedral", "slug": "binary-octahedral", "name": "Binary octahedral group", "symbol": "2O", "category": "finite", "level": "reference", "aliases": [ "CSU(2,3)", "Binary octahedral", "2O" ], "searchTerms": [ "Binary octahedral group", "2O", "CSU(2,3)", "Binary octahedral", "2O", "SmallGroup(48,28)" ], "tags": [ "Finite reference", "Nonabelian" ], "summary": "The binary rotation group of the cube or octahedron, a central double cover of S₄.", "definition": "The inverse image of the octahedral rotation group in SU(2).", "definitionEvidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases" } ], "properties": [ { "key": "order", "label": "Order", "value": "48", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 2, 2, 2, 3, 3, 4", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(48,28)", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "The faithful doublets and the four-dimensional irrep are quaternionic. The other irreps descend to the rotational quotient S₄.", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:binary-polyhedral", "label": "Family", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Heading; Aliases" } ] }, { "type": "central-quotient", "target": "group:s-4", "label": "2O / C₂ ≅ S₄", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Centre: G/Z" } ] }, { "type": "subgroup-of", "target": "group:su-2", "label": "Binary octahedral embedding", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "48", "abelian": false, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6", "ρ7", "ρ8" ], "dimensions": [ 1, 1, 2, 2, 2, 3, 3, 4 ], "reality": [ "Real", "Real", "Real", "Quaternionic", "Quaternionic", "Real", "Real", "Quaternionic" ] } } }, { "id": "group:binary-icosahedral", "slug": "binary-icosahedral", "name": "Binary icosahedral group", "symbol": "2I", "category": "finite", "level": "reference", "aliases": [ "SL(2,5)", "2.A5", "Binary icosahedral" ], "searchTerms": [ "Binary icosahedral group", "2I", "SL(2,5)", "2.A5", "Binary icosahedral", "SmallGroup(120,5)" ], "tags": [ "Finite reference", "Nonabelian" ], "summary": "A perfect central double cover of A₅; it is quasisimple, with a nontrivial center.", "definition": "The inverse image of the icosahedral rotation group in SU(2); abstractly SL(2,5).", "definitionEvidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases" } ], "properties": [ { "key": "order", "label": "Order", "value": "120", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 2, 2, 3, 3, 4, 4, 5, 6", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(120,5)", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "The two four-dimensional irreps differ: one is real and descends to A₅; the other is quaternionic and faithful.", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:binary-polyhedral", "label": "Family", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Heading; Aliases" } ] }, { "type": "central-quotient", "target": "group:a-5", "label": "2I / C₂ ≅ A₅", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Centre: G/Z" } ] }, { "type": "subgroup-of", "target": "group:su-2", "label": "Binary icosahedral embedding", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "120", "abelian": false, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6", "ρ7", "ρ8", "ρ9" ], "dimensions": [ 1, 2, 2, 3, 3, 4, 4, 5, 6 ], "reality": [ "Real", "Quaternionic", "Quaternionic", "Real", "Real", "Real", "Quaternionic", "Real", "Quaternionic" ] } } }, { "id": "group:psl-2-7", "slug": "psl-2-7", "name": "Projective special linear group PSL(2,7)", "symbol": "PSL(2,7)", "category": "finite", "level": "reference", "aliases": [ "PSL(3,2)", "GL(3,2)", "SL(3,2)", "L2(7)", "L3(2)" ], "searchTerms": [ "Projective special linear group PSL(2,7)", "PSL(2,7)", "PSL(3,2)", "GL(3,2)", "SL(3,2)", "L2(7)", "L3(2)", "SmallGroup(168,42)" ], "tags": [ "Finite reference", "Finite simple", "Nonabelian" ], "summary": "A finite simple group of order 168 with a conjugate pair of complex triplets.", "definition": "SL(2,F₇)/{±I}; equivalently GL(3,F₂).", "definitionEvidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Heading; Aliases; Generators and relations" } ], "properties": [ { "key": "order", "label": "Order", "value": "168", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Simple label" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 3, 3, 6, 7, 8", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(168,42)", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "The two triplets are inequivalent complex conjugates. The 6-, 7- and 8-dimensional irreps are real.", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-linear", "label": "Family", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Heading; Aliases" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "168", "abelian": false, "simple": true, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6" ], "dimensions": [ 1, 3, 3, 6, 7, 8 ], "reality": [ "Real", "Complex", "Complex", "Real", "Real", "Real" ] } } }, { "id": "group:delta-27", "slug": "delta-27", "name": "Delta(27), the finite Heisenberg group", "symbol": "Δ(27)", "category": "finite", "level": "reference", "aliases": [ "Delta(27)", "He3", "Heisenberg group of order 27", "3_+^(1+2)", "(C3 × C3) ⋊ C3" ], "searchTerms": [ "Delta(27), the finite Heisenberg group", "Δ(27)", "Delta(27)", "He3", "Heisenberg group of order 27", "3_+^(1+2)", "(C3 × C3) ⋊ C3", "SmallGroup(27,3)" ], "tags": [ "Finite reference", "Nonabelian" ], "summary": "The nonabelian Heisenberg group over F₃, with nine singlets and two conjugate triplets.", "definition": "The order-27 subgroup of SU(3) generated by diag(ω,1,ω⁻¹), diag(1,ω,ω⁻¹) and the cyclic coordinate permutation, with ω³ = 1 and ω ≠ 1.", "definitionEvidence": [ { "source": "finite-benvenuti-delta27", "locator": "Equation (3.12)" }, { "source": "finite-groupnames-delta-27", "locator": "Heading and Aliases" } ], "properties": [ { "key": "order", "label": "Order", "value": "27", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₃", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Chief series; Centre" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(27,3)", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "The center and commutator subgroup are both C₃; the abelianization is C₃ × C₃.", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Centre; Commutator" } ], "assumptions": [] }, { "text": "The SU(3) matrix realization is used in the cited study of invariant rings and orbifold gauge theories.", "evidence": [ { "source": "finite-benvenuti-delta27", "locator": "Section 3, equation (3.12) and following discussion" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "subgroup-of", "target": "group:su-3", "label": "Specified SU(3) embedding", "evidence": [ { "source": "finite-benvenuti-delta27", "locator": "Equation (3.12)" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "27", "abelian": false, "simple": false, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6", "ρ7", "ρ8", "ρ9", "ρ10", "ρ11" ], "dimensions": [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3 ], "reality": [ "Real", "Complex", "Complex", "Complex", "Complex", "Complex", "Complex", "Complex", "Complex", "Complex", "Complex" ] } } }, { "id": "group:a-6", "slug": "a-6", "name": "Alternating group on six letters", "symbol": "A₆", "category": "finite", "level": "reference", "aliases": [ "Alt(6)", "PSL(2,9)", "L2(9)", "PSp(4,2) derived subgroup" ], "searchTerms": [ "Alternating group on six letters", "A₆", "Alt(6)", "PSL(2,9)", "L2(9)", "PSp(4,2) derived subgroup", "SmallGroup(360,118)" ], "tags": [ "Finite reference", "Finite simple", "Nonabelian" ], "summary": "A finite simple group in both the alternating and projective linear classification families.", "definition": "The group of even permutations of six letters; also PSL(2,9).", "definitionEvidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading and Aliases" }, { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "properties": [ { "key": "order", "label": "Order", "value": "360", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading: order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Centre" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Commutator; character table" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Simple label" } ], "assumptions": [] }, { "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 5, 5, 8, 8, 9, 10", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ] }, { "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(360,118)", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Representation scope", "paragraphs": [ { "text": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Character table" } ], "assumptions": [] }, { "text": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [] } ] }, { "id": "structure", "title": "Structure and conventions", "paragraphs": [ { "text": "Both five-dimensional irreps and both eight-dimensional irreps are distinct; dimension alone does not label an irrep.", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-alternating", "label": "Family", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading; Aliases" } ] }, { "type": "member-of", "target": "family:finite-linear", "label": "Family", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading; Aliases" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "360", "abelian": false, "simple": true, "irreps": { "category": "ordinary-complex", "complete": true, "labels": [ "ρ1", "ρ2", "ρ3", "ρ4", "ρ5", "ρ6", "ρ7" ], "dimensions": [ 1, 5, 5, 8, 8, 9, 10 ], "reality": [ "Real", "Real", "Real", "Real", "Real", "Real", "Real" ] } } }, { "id": "group:sporadic-m11", "slug": "sporadic-m11", "name": "Mathieu group M11", "symbol": "M₁₁", "category": "finite", "level": "classification", "aliases": [ "M11" ], "searchTerms": [ "Mathieu group M11", "M₁₁", "M11" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 7920.", "definition": "The sporadic simple group M₁₁ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-m11", "locator": "Heading and Order" }, { "source": "finite-atlas-m11", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "M₁₁ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "7920", "evidence": [ { "source": "finite-atlas-m11", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₁₁ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-m11", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-m11", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₁₁ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "7920", "abelian": false, "simple": true } }, { "id": "group:sporadic-m12", "slug": "sporadic-m12", "name": "Mathieu group M12", "symbol": "M₁₂", "category": "finite", "level": "classification", "aliases": [ "M12" ], "searchTerms": [ "Mathieu group M12", "M₁₂", "M12" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 95040.", "definition": "The sporadic simple group M₁₂ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-m12", "locator": "Heading and Order" }, { "source": "finite-atlas-m12", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "M₁₂ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "95040", "evidence": [ { "source": "finite-atlas-m12", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₁₂ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-m12", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-m12", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₁₂ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "95040", "abelian": false, "simple": true } }, { "id": "group:sporadic-m22", "slug": "sporadic-m22", "name": "Mathieu group M22", "symbol": "M₂₂", "category": "finite", "level": "classification", "aliases": [ "M22" ], "searchTerms": [ "Mathieu group M22", "M₂₂", "M22" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 443520.", "definition": "The sporadic simple group M₂₂ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-m22", "locator": "Heading and Order" }, { "source": "finite-atlas-m22", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "M₂₂ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "443520", "evidence": [ { "source": "finite-atlas-m22", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₂₂ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-m22", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-m22", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₂₂ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "443520", "abelian": false, "simple": true } }, { "id": "group:sporadic-m23", "slug": "sporadic-m23", "name": "Mathieu group M23", "symbol": "M₂₃", "category": "finite", "level": "classification", "aliases": [ "M23" ], "searchTerms": [ "Mathieu group M23", "M₂₃", "M23" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 10200960.", "definition": "The sporadic simple group M₂₃ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-m23", "locator": "Heading and Order" }, { "source": "finite-atlas-m23", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "M₂₃ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "10200960", "evidence": [ { "source": "finite-atlas-m23", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₂₃ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-m23", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-m23", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₂₃ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "10200960", "abelian": false, "simple": true } }, { "id": "group:sporadic-m24", "slug": "sporadic-m24", "name": "Mathieu group M24", "symbol": "M₂₄", "category": "finite", "level": "classification", "aliases": [ "M24" ], "searchTerms": [ "Mathieu group M24", "M₂₄", "M24" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 244823040.", "definition": "The sporadic simple group M₂₄ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-m24", "locator": "Heading and Order" }, { "source": "finite-atlas-m24", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "M₂₄ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "244823040", "evidence": [ { "source": "finite-atlas-m24", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₂₄ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-m24", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-m24", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₂₄ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "244823040", "abelian": false, "simple": true } }, { "id": "group:sporadic-j1", "slug": "sporadic-j1", "name": "First Janko group", "symbol": "J₁", "category": "finite", "level": "classification", "aliases": [ "J1" ], "searchTerms": [ "First Janko group", "J₁", "J1" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 175560.", "definition": "The sporadic simple group J₁ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-j1", "locator": "Heading and Order" }, { "source": "finite-atlas-j1", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "J₁ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "175560", "evidence": [ { "source": "finite-atlas-j1", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "J₁ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-j1", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-j1", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "J₁ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "175560", "abelian": false, "simple": true } }, { "id": "group:sporadic-j2", "slug": "sporadic-j2", "name": "Hall–Janko group", "symbol": "J₂", "category": "finite", "level": "classification", "aliases": [ "J2", "Hall-Janko" ], "searchTerms": [ "Hall–Janko group", "J₂", "J2", "Hall-Janko" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 604800.", "definition": "The sporadic simple group J₂ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-j2", "locator": "Heading and Order" }, { "source": "finite-atlas-j2", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "J₂ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "604800", "evidence": [ { "source": "finite-atlas-j2", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "J₂ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-j2", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-j2", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "J₂ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "604800", "abelian": false, "simple": true } }, { "id": "group:sporadic-j3", "slug": "sporadic-j3", "name": "Third Janko group", "symbol": "J₃", "category": "finite", "level": "classification", "aliases": [ "J3" ], "searchTerms": [ "Third Janko group", "J₃", "J3" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 50232960.", "definition": "The sporadic simple group J₃ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-j3", "locator": "Heading and Order" }, { "source": "finite-atlas-j3", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "J₃ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "50232960", "evidence": [ { "source": "finite-atlas-j3", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "J₃ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-j3", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-j3", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "J₃ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "50232960", "abelian": false, "simple": true } }, { "id": "group:sporadic-j4", "slug": "sporadic-j4", "name": "Fourth Janko group", "symbol": "J₄", "category": "finite", "level": "classification", "aliases": [ "J4" ], "searchTerms": [ "Fourth Janko group", "J₄", "J4" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 86775571046077562880.", "definition": "The sporadic simple group J₄ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-j4", "locator": "Heading and Order" }, { "source": "finite-atlas-j4", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "J₄ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "86775571046077562880", "evidence": [ { "source": "finite-atlas-j4", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "J₄ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-j4", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-j4", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "J₄ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "86775571046077562880", "abelian": false, "simple": true } }, { "id": "group:sporadic-co1", "slug": "sporadic-co1", "name": "First Conway group", "symbol": "Co₁", "category": "finite", "level": "classification", "aliases": [ "Co1" ], "searchTerms": [ "First Conway group", "Co₁", "Co1" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 4157776806543360000.", "definition": "The sporadic simple group Co₁ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-co1", "locator": "Heading and Order" }, { "source": "finite-atlas-co1", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Co₁ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "4157776806543360000", "evidence": [ { "source": "finite-atlas-co1", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Co₁ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-co1", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-co1", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Co₁ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "4157776806543360000", "abelian": false, "simple": true } }, { "id": "group:sporadic-co2", "slug": "sporadic-co2", "name": "Second Conway group", "symbol": "Co₂", "category": "finite", "level": "classification", "aliases": [ "Co2" ], "searchTerms": [ "Second Conway group", "Co₂", "Co2" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 42305421312000.", "definition": "The sporadic simple group Co₂ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-co2", "locator": "Heading and Order" }, { "source": "finite-atlas-co2", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Co₂ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "42305421312000", "evidence": [ { "source": "finite-atlas-co2", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Co₂ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-co2", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-co2", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Co₂ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "42305421312000", "abelian": false, "simple": true } }, { "id": "group:sporadic-co3", "slug": "sporadic-co3", "name": "Third Conway group", "symbol": "Co₃", "category": "finite", "level": "classification", "aliases": [ "Co3" ], "searchTerms": [ "Third Conway group", "Co₃", "Co3" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 495766656000.", "definition": "The sporadic simple group Co₃ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-co3", "locator": "Heading and Order" }, { "source": "finite-atlas-co3", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Co₃ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "495766656000", "evidence": [ { "source": "finite-atlas-co3", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Co₃ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-co3", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-co3", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Co₃ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "495766656000", "abelian": false, "simple": true } }, { "id": "group:sporadic-fi22", "slug": "sporadic-fi22", "name": "Fischer group Fi22", "symbol": "Fi₂₂", "category": "finite", "level": "classification", "aliases": [ "Fi22", "F22" ], "searchTerms": [ "Fischer group Fi22", "Fi₂₂", "Fi22", "F22" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 64561751654400.", "definition": "The sporadic simple group Fi₂₂ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-fi22", "locator": "Heading and Order" }, { "source": "finite-atlas-fi22", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₂ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "64561751654400", "evidence": [ { "source": "finite-atlas-fi22", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₂ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-fi22", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-fi22", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₂ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "64561751654400", "abelian": false, "simple": true } }, { "id": "group:sporadic-fi23", "slug": "sporadic-fi23", "name": "Fischer group Fi23", "symbol": "Fi₂₃", "category": "finite", "level": "classification", "aliases": [ "Fi23", "F23" ], "searchTerms": [ "Fischer group Fi23", "Fi₂₃", "Fi23", "F23" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 4089470473293004800.", "definition": "The sporadic simple group Fi₂₃ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-fi23", "locator": "Heading and Order" }, { "source": "finite-atlas-fi23", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₃ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "4089470473293004800", "evidence": [ { "source": "finite-atlas-fi23", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₃ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-fi23", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-fi23", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₃ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "4089470473293004800", "abelian": false, "simple": true } }, { "id": "group:sporadic-fi24-prime", "slug": "sporadic-fi24-prime", "name": "Fischer simple group Fi24 prime", "symbol": "Fi₂₄′", "category": "finite", "level": "classification", "aliases": [ "Fi24'", "Fi24 prime", "F24" ], "searchTerms": [ "Fischer simple group Fi24 prime", "Fi₂₄′", "Fi24'", "Fi24 prime", "F24" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 1255205709190661721292800.", "definition": "The sporadic simple group Fi₂₄′ identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-fi24-prime", "locator": "Heading and Order" }, { "source": "finite-atlas-fi24-prime", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₄′ entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "1255205709190661721292800", "evidence": [ { "source": "finite-atlas-fi24-prime", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₄′ entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-fi24-prime", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-fi24-prime", "locator": "Representations and representation headings" } ], "assumptions": [] }, { "text": "The prime in Fi₂₄′ is part of the simple-group name. Fi₂₄′.2 is a larger extension.", "evidence": [ { "source": "finite-atlas-fi24-prime", "locator": "Group name and automorphism-group data" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Fi₂₄′ entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "1255205709190661721292800", "abelian": false, "simple": true } }, { "id": "group:sporadic-hs", "slug": "sporadic-hs", "name": "Higman–Sims group", "symbol": "HS", "category": "finite", "level": "classification", "aliases": [ "Higman-Sims" ], "searchTerms": [ "Higman–Sims group", "HS", "Higman-Sims" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 44352000.", "definition": "The sporadic simple group HS identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-hs", "locator": "Heading and Order" }, { "source": "finite-atlas-hs", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "HS entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "44352000", "evidence": [ { "source": "finite-atlas-hs", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "HS entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-hs", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-hs", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "HS entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "44352000", "abelian": false, "simple": true } }, { "id": "group:sporadic-mcl", "slug": "sporadic-mcl", "name": "McLaughlin group", "symbol": "McL", "category": "finite", "level": "classification", "aliases": [ "McL" ], "searchTerms": [ "McLaughlin group", "McL", "McL" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 898128000.", "definition": "The sporadic simple group McL identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-mcl", "locator": "Heading and Order" }, { "source": "finite-atlas-mcl", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "McL entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "898128000", "evidence": [ { "source": "finite-atlas-mcl", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "McL entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-mcl", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-mcl", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "McL entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "898128000", "abelian": false, "simple": true } }, { "id": "group:sporadic-he", "slug": "sporadic-he", "name": "Held group", "symbol": "He", "category": "finite", "level": "classification", "aliases": [ "Held" ], "searchTerms": [ "Held group", "He", "Held" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 4030387200.", "definition": "The sporadic simple group He identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-he", "locator": "Heading and Order" }, { "source": "finite-atlas-he", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "He entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "4030387200", "evidence": [ { "source": "finite-atlas-he", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "He entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-he", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-he", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "He entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "4030387200", "abelian": false, "simple": true } }, { "id": "group:sporadic-ru", "slug": "sporadic-ru", "name": "Rudvalis group", "symbol": "Ru", "category": "finite", "level": "classification", "aliases": [ "Rudvalis" ], "searchTerms": [ "Rudvalis group", "Ru", "Rudvalis" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 145926144000.", "definition": "The sporadic simple group Ru identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-ru", "locator": "Heading and Order" }, { "source": "finite-atlas-ru", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Ru entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "145926144000", "evidence": [ { "source": "finite-atlas-ru", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Ru entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-ru", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-ru", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Ru entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "145926144000", "abelian": false, "simple": true } }, { "id": "group:sporadic-suz", "slug": "sporadic-suz", "name": "Sporadic Suzuki group", "symbol": "Suz", "category": "finite", "level": "classification", "aliases": [ "Suzuki sporadic" ], "searchTerms": [ "Sporadic Suzuki group", "Suz", "Suzuki sporadic" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 448345497600.", "definition": "The sporadic simple group Suz identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-suz", "locator": "Heading and Order" }, { "source": "finite-atlas-suz", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Suz entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "448345497600", "evidence": [ { "source": "finite-atlas-suz", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Suz entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-suz", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-suz", "locator": "Representations and representation headings" } ], "assumptions": [] }, { "text": "This isolated sporadic group is distinct from the infinite Suzuki family Sz(q) = ²B₂(q).", "evidence": [ { "source": "finite-atlas-suz", "locator": "Heading" }, { "source": "finite-wilson-exceptional", "locator": "Suzuki families" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Suz entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "448345497600", "abelian": false, "simple": true } }, { "id": "group:sporadic-on", "slug": "sporadic-on", "name": "O’Nan group", "symbol": "O’N", "category": "finite", "level": "classification", "aliases": [ "O'Nan", "O'N", "ON" ], "searchTerms": [ "O’Nan group", "O’N", "O'Nan", "O'N", "ON" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 460815505920.", "definition": "The sporadic simple group O’N identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-on", "locator": "Heading and Order" }, { "source": "finite-atlas-on", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "O’N entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "460815505920", "evidence": [ { "source": "finite-atlas-on", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "O’N entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-on", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-on", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "O’N entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "460815505920", "abelian": false, "simple": true } }, { "id": "group:sporadic-hn", "slug": "sporadic-hn", "name": "Harada–Norton group", "symbol": "HN", "category": "finite", "level": "classification", "aliases": [ "Harada-Norton" ], "searchTerms": [ "Harada–Norton group", "HN", "Harada-Norton" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 273030912000000.", "definition": "The sporadic simple group HN identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-hn", "locator": "Heading and Order" }, { "source": "finite-atlas-hn", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "HN entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "273030912000000", "evidence": [ { "source": "finite-atlas-hn", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "HN entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-hn", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-hn", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "HN entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "273030912000000", "abelian": false, "simple": true } }, { "id": "group:sporadic-ly", "slug": "sporadic-ly", "name": "Lyons group", "symbol": "Ly", "category": "finite", "level": "classification", "aliases": [ "Lyons" ], "searchTerms": [ "Lyons group", "Ly", "Lyons" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 51765179004000000.", "definition": "The sporadic simple group Ly identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-ly", "locator": "Heading and Order" }, { "source": "finite-atlas-ly", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Ly entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "51765179004000000", "evidence": [ { "source": "finite-atlas-ly", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Ly entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-ly", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-ly", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Ly entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "51765179004000000", "abelian": false, "simple": true } }, { "id": "group:sporadic-th", "slug": "sporadic-th", "name": "Thompson group", "symbol": "Th", "category": "finite", "level": "classification", "aliases": [ "Thompson sporadic" ], "searchTerms": [ "Thompson group", "Th", "Thompson sporadic" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 90745943887872000.", "definition": "The sporadic simple group Th identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-th", "locator": "Heading and Order" }, { "source": "finite-atlas-th", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "Th entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "90745943887872000", "evidence": [ { "source": "finite-atlas-th", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Th entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-th", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-th", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Th entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "90745943887872000", "abelian": false, "simple": true } }, { "id": "group:sporadic-b", "slug": "sporadic-b", "name": "Baby Monster group", "symbol": "B", "category": "finite", "level": "classification", "aliases": [ "Baby Monster", "BM" ], "searchTerms": [ "Baby Monster group", "B", "Baby Monster", "BM" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 4154781481226426191177580544000000.", "definition": "The sporadic simple group B identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-b", "locator": "Heading and Order" }, { "source": "finite-atlas-b", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "B entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "4154781481226426191177580544000000", "evidence": [ { "source": "finite-atlas-b", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "B entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-b", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-b", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "B entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "4154781481226426191177580544000000", "abelian": false, "simple": true } }, { "id": "group:sporadic-m", "slug": "sporadic-m", "name": "Monster group", "symbol": "M", "category": "finite", "level": "classification", "aliases": [ "Monster", "F1" ], "searchTerms": [ "Monster group", "M", "Monster", "F1" ], "tags": [ "Finite simple", "Sporadic", "Classification" ], "summary": "One of the 26 sporadic finite simple groups; order 808017424794512875886459904961710757005754368000000000.", "definition": "The sporadic simple group M identified by the ATLAS standard generators and group record.", "definitionEvidence": [ { "source": "finite-atlas-m", "locator": "Heading and Order" }, { "source": "finite-atlas-m", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "M entry" } ], "properties": [ { "key": "order", "label": "Order", "value": "808017424794512875886459904961710757005754368000000000", "evidence": [ { "source": "finite-atlas-m", "locator": "Order" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M entry in sporadic simple groups" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [] }, { "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-m", "locator": "Representations" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Reference scope", "paragraphs": [ { "text": "This entry establishes the group identity and exact order. Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-m", "locator": "Representations and representation headings" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:finite-sporadic", "label": "Sporadic classification", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M entry" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "808017424794512875886459904961710757005754368000000000", "abelian": false, "simple": true } }, { "id": "group:tits", "slug": "tits", "name": "Tits group", "symbol": "²F₄(2)′", "category": "finite", "level": "classification", "aliases": [ "2F4(2)'", "Tits", "T" ], "searchTerms": [ "Tits group", "Ree", "2F4(2) derived subgroup" ], "tags": [ "Finite simple", "Lie-type exception", "Classification" ], "summary": "The simple derived subgroup of ²F₄(2), of index two and order 17971200.", "definition": "The commutator subgroup ²F₄(2)′ of the smallest large Ree group.", "definitionEvidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree small cases" }, { "source": "finite-atlas-tits", "locator": "Heading and standard generators" } ], "properties": [ { "key": "order", "label": "Order", "value": "17971200", "evidence": [ { "source": "finite-atlas-tits", "locator": "Order" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "finite-atlas-tits", "locator": "Simple group identity" } ], "derivation": { "method": "The center of a nonabelian simple group is trivial.", "inputs": [ "group:tits#simple", "group:tits#abelian" ] }, "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree small cases" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Simple Tits group of composite order" } ], "assumptions": [] }, { "key": "classification-convention", "label": "Classification convention", "value": "Lie-type small case; excluded from the 26 sporadic groups.", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Miscellaneous" }, { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree small cases" } ], "assumptions": [] } ], "sections": [ { "id": "scope", "title": "Identity and representation scope", "paragraphs": [ { "text": "²F₄(2) itself is not simple. Its simple derived subgroup is the Tits group, and the parent group is its extension by an outer involution.", "evidence": [ { "source": "finite-atlas-tits", "locator": "Standard generators of ²F₄(2) = T.2" } ], "assumptions": [] }, { "text": "This classification record links to available ATLAS representations. It does not assert a complete local list of ordinary complex irreducibles.", "evidence": [ { "source": "finite-atlas-tits", "locator": "Representations" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "exceptional-member-of", "target": "family:finite-ree-f4", "label": "Derived small case of ²F₄", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree small cases" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "finite", "order": "17971200", "abelian": false, "simple": true } }, { "id": "family:lie-a", "slug": "lie-a", "name": "Simple Lie type Aᵣ", "symbol": "Aᵣ", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "Ar", "Dynkin classification", "simple Lie algebra", "SU(r + 1)" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type Aᵣ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "sl(r + 1, C), with integer r ≥ 1.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "integer r ≥ 1", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "r(r + 2)", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "r + 1", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "r + 1", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "SU(r + 1)", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "Cᵣ₊₁", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Classical real forms", "paragraphs": [ { "text": "For n = r + 1 the real forms are su(p,q), p + q = n; sl(n,R); and su*(2m) = sl(m,H) when n = 2m, with low-rank coincidences understood.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences" } ], "assumptions": [] } ] }, { "id": "low-rank", "title": "Rank conventions and coincidences", "paragraphs": [ { "text": "r = 1: A₁ = B₁ = C₁. The canonical family entry is A.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:a-1-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] }, { "type": "has-example", "target": "algebra:a-2-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] }, { "type": "has-example", "target": "algebra:a-3-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] }, { "type": "has-example", "target": "algebra:a-4-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "r", "domain": "integer r ≥ 1" } ], "exceptions": [ { "at": "r = 1", "description": "A₁ = B₁ = C₁. The canonical family entry is A." } ], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "family:lie-b", "slug": "lie-b", "name": "Simple Lie type Bᵣ", "symbol": "Bᵣ", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "Br", "Dynkin classification", "simple Lie algebra", "Spin(2r + 1)" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type Bᵣ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "so(2r + 1, C), with integer r ≥ 2.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "integer r ≥ 2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "r(2r + 1)", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "2r", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "2r − 1", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "Spin(2r + 1)", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Classical real forms", "paragraphs": [ { "text": "The real forms are orthogonal algebras so(p,q) with p + q = 2r + 1. Interchanging p and q gives the same real algebra.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences" } ], "assumptions": [] } ] }, { "id": "low-rank", "title": "Rank conventions and coincidences", "paragraphs": [ { "text": "r = 1: B₁ = A₁, represented in family A.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] }, { "text": "r = 2: B₂ = C₂ as complex Lie algebras. Root-length exchange changes the conventional Dynkin-node labeling.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:b-2-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "r", "domain": "integer r ≥ 2" } ], "exceptions": [ { "at": "r = 1", "description": "B₁ = A₁, represented in family A." }, { "at": "r = 2", "description": "B₂ = C₂ as complex Lie algebras. Root-length exchange changes the conventional Dynkin-node labeling." } ], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "family:lie-c", "slug": "lie-c", "name": "Simple Lie type Cᵣ", "symbol": "Cᵣ", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "Cr", "Dynkin classification", "simple Lie algebra", "Sp(r), compact quaternionic convention" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type Cᵣ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "sp(2r, C), with integer r ≥ 3 in the nonredundant classification.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "integer r ≥ 3 in the nonredundant classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "r(2r + 1)", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "2r", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "r + 1", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "Sp(r), compact quaternionic convention", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Classical real forms", "paragraphs": [ { "text": "The real forms are sp(2r,R) and sp(p,q), p + q = r. Here sp(p,q) preserves a quaternionic Hermitian form, whereas sp(2r,R) acts on a real vector space of dimension 2r.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences" } ], "assumptions": [] } ] }, { "id": "low-rank", "title": "Rank conventions and coincidences", "paragraphs": [ { "text": "r = 1: C₁ = A₁.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] }, { "text": "r = 2: C₂ = B₂; see the canonical B₂ algebra and compact Sp(2) group records.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:c-3-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "r", "domain": "integer r ≥ 3 in the nonredundant classification" } ], "exceptions": [ { "at": "r = 1", "description": "C₁ = A₁." }, { "at": "r = 2", "description": "C₂ = B₂; see the canonical B₂ algebra and compact Sp(2) group records." } ], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "family:lie-d", "slug": "lie-d", "name": "Simple Lie type Dᵣ", "symbol": "Dᵣ", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "Dr", "Dynkin classification", "simple Lie algebra", "Spin(2r)" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type Dᵣ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "so(2r, C), with integer r ≥ 4.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "integer r ≥ 4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "r(2r − 1)", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "2r − 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "2r − 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "Spin(2r)", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "C₄ for r odd; C₂ × C₂ for r even", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Classical real forms", "paragraphs": [ { "text": "The real forms are so(p,q), p + q = 2r, and so*(2r). There are low-rank coincidences, including so*(8) ≅ so(6,2).", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences" } ], "assumptions": [] } ] }, { "id": "low-rank", "title": "Rank conventions and coincidences", "paragraphs": [ { "text": "r = 1: so(2,C) is abelian, not simple.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] }, { "text": "r = 2: D₂ = A₁ ⊕ A₁ is semisimple, not simple.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] }, { "text": "r = 3: D₃ = A₃; its compact simply connected group is Spin(6) ≅ SU(4).", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] }, { "text": "r = 4: Triality permutes the three nonidentity central elements of Spin(8), and hence the three order-two quotient kernels.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:d-4-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] }, { "type": "has-example", "target": "algebra:d-5-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [ { "name": "r", "domain": "integer r ≥ 4" } ], "exceptions": [ { "at": "r = 1", "description": "so(2,C) is abelian, not simple." }, { "at": "r = 2", "description": "D₂ = A₁ ⊕ A₁ is semisimple, not simple." }, { "at": "r = 3", "description": "D₃ = A₃; its compact simply connected group is Spin(6) ≅ SU(4)." }, { "at": "r = 4", "description": "Triality permutes the three nonidentity central elements of Spin(8), and hence the three order-two quotient kernels." } ], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "family:lie-e6", "slug": "lie-e6", "name": "Simple Lie type E₆", "symbol": "E₆", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "E₆", "Dynkin classification", "simple Lie algebra", "E₆, compact simply connected form" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type E₆ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "The simple complex Lie algebra with Dynkin diagram E₆.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "6", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "78", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "12", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "12", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "E₆, compact simply connected form", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "C₃", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Current real-form coverage", "paragraphs": [ { "text": "This foundation supplies the compact real form of this exceptional type. It does not yet provide the complete list of exceptional noncompact real forms or their global covers.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:e-6-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [], "exceptions": [], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "family:lie-e7", "slug": "lie-e7", "name": "Simple Lie type E₇", "symbol": "E₇", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "E₇", "Dynkin classification", "simple Lie algebra", "E₇, compact simply connected form" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type E₇ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "The simple complex Lie algebra with Dynkin diagram E₇.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "7", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "133", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "18", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "18", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "E₇, compact simply connected form", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. 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Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Current real-form coverage", "paragraphs": [ { "text": "This foundation supplies the compact real form of this exceptional type. It does not yet provide the complete list of exceptional noncompact real forms or their global covers.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:e-7-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [], "exceptions": [], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "family:lie-e8", "slug": "lie-e8", "name": "Simple Lie type E₈", "symbol": "E₈", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "E₈", "Dynkin classification", "simple Lie algebra", "E₈, compact form (simply connected and adjoint)" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type E₈ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "The simple complex Lie algebra with Dynkin diagram E₈.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "8", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "248", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "30", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "30", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "E₈, compact form (simply connected and adjoint)", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Current real-form coverage", "paragraphs": [ { "text": "This foundation supplies the compact real form of this exceptional type. It does not yet provide the complete list of exceptional noncompact real forms or their global covers.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:e-8-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [], "exceptions": [], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "family:lie-f4", "slug": "lie-f4", "name": "Simple Lie type F₄", "symbol": "F₄", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "F₄", "Dynkin classification", "simple Lie algebra", "F₄, compact form (simply connected and adjoint)" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type F₄ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "The simple complex Lie algebra with Dynkin diagram F₄.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "52", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "12", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "9", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "F₄, compact form (simply connected and adjoint)", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Current real-form coverage", "paragraphs": [ { "text": "This foundation supplies the compact real form of this exceptional type. It does not yet provide the complete list of exceptional noncompact real forms or their global covers.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:f-4-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [], "exceptions": [], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "family:lie-g2", "slug": "lie-g2", "name": "Simple Lie type G₂", "symbol": "G₂", "category": "lie", "level": "classification", "aliases": [], "searchTerms": [ "G₂", "Dynkin classification", "simple Lie algebra", "G₂, compact form (simply connected and adjoint)" ], "tags": [ "Classification", "Complex simple algebra", "Compact global forms" ], "summary": "Complex simple Lie type G₂ with dimension, normalized Coxeter data, compact cover, and its center.", "definition": "The simple complex Lie algebra with Dynkin diagram G₂.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "rank", "label": "Rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex algebra dimension", "value": "14", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "derivation": { "method": "For an irreducible root system, dim(g) = r + |Φ| = r(h + 1).", "inputs": [ "Rank r", "Coxeter number h in Section 32.4" ] }, "assumptions": [] }, { "key": "coxeter-number", "label": "Coxeter number h", "value": "6", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "dual-coxeter-number", "label": "Dual Coxeter number h∨", "value": "4", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "compact-cover", "label": "Compact simply connected group", "value": "G₂, compact form (simply connected and adjoint)", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "center", "label": "Center of compact simply connected group", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "root-normalization", "label": "Root-length convention", "value": "Long roots have squared length 2", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ], "sections": [ { "id": "objects", "title": "Algebra, real form, global group", "paragraphs": [ { "text": "A Dynkin type classifies a complex simple Lie algebra. A real Lie group additionally requires a real form and a global form. The compact cover in this record is a specific connected group, not the meaning of the type symbol itself.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Connected compact groups with this simple Lie algebra are quotients of the simply connected compact group by subgroups of its center. Distinct central subgroups need not yield distinct abstract groups: automorphisms of the covering group can identify the resulting quotients.", "evidence": [ { "source": "conrad", "locator": "Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation convention", "paragraphs": [ { "text": "Dominant integral highest weights classify finite-dimensional complex irreducible representations of the simply connected compact group. A representation descends to a chosen central quotient precisely when every element of the quotient kernel acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The listed dual Coxeter number uses long-root squared length 2. Relating it to a physics Casimir convention requires the normalization of generators and invariant bilinear form.", "evidence": [ { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] } ] }, { "id": "real-forms", "title": "Current real-form coverage", "paragraphs": [ { "text": "This foundation supplies the compact real form of this exceptional type. It does not yet provide the complete list of exceptional noncompact real forms or their global covers.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-example", "target": "algebra:g-2-c", "label": "Concrete algebra record", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "family", "data": { "parameters": [], "exceptions": [], "scope": "Classification of complex simple type; compact center and global-form rule; selected concrete real forms." } }, { "id": "algebra:a-1-c", "slug": "a-1-c", "name": "sl(2, C) — complex Lie algebra", "symbol": "sl(2, C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "A₁", "sl(2, C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex traceless 2 × 2 matrices with commutator bracket.", "definition": "The complex traceless 2 × 2 matrices with commutator bracket.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "1", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "A₁", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-a", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 3, "rank": 1, "cartanType": "A₁", "simple": true } }, { "id": "algebra:a-2-c", "slug": "a-2-c", "name": "sl(3, C) — complex Lie algebra", "symbol": "sl(3, C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "A₂", "sl(3, C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex traceless 3 × 3 matrices with commutator bracket.", "definition": "The complex traceless 3 × 3 matrices with commutator bracket.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "8", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "A₂", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-a", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 8, "rank": 2, "cartanType": "A₂", "simple": true } }, { "id": "algebra:a-3-c", "slug": "a-3-c", "name": "sl(4, C) — complex Lie algebra", "symbol": "sl(4, C)", "category": "lie", "level": "reference", "aliases": [ "so(6,C)", "D₃" ], "searchTerms": [ "A₃", "sl(4, C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex traceless 4 × 4 matrices with commutator bracket; isomorphic to so(6,C).", "definition": "The complex traceless 4 × 4 matrices with commutator bracket; isomorphic to so(6,C).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "A₃", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-a", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 15, "rank": 3, "cartanType": "A₃", "simple": true } }, { "id": "algebra:a-4-c", "slug": "a-4-c", "name": "sl(5, C) — complex Lie algebra", "symbol": "sl(5, C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "A₄", "sl(5, C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex traceless 5 × 5 matrices with commutator bracket.", "definition": "The complex traceless 5 × 5 matrices with commutator bracket.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "24", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "A₄", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-a", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 24, "rank": 4, "cartanType": "A₄", "simple": true } }, { "id": "algebra:b-2-c", "slug": "b-2-c", "name": "so(5, C) ≅ sp(4, C) — complex Lie algebra", "symbol": "so(5, C) ≅ sp(4, C)", "category": "lie", "level": "reference", "aliases": [ "sp(4,C)", "C₂" ], "searchTerms": [ "B₂ = C₂", "so(5, C) ≅ sp(4, C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex simple algebra of type B₂, isomorphic to the symplectic Lie algebra on C⁴. The B₂ and C₂ diagrams use different conventional node numberings.", "definition": "The complex simple algebra of type B₂, isomorphic to the symplectic Lie algebra on C⁴. The B₂ and C₂ diagrams use different conventional node numberings.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "B₂ = C₂", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-b", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] }, { "type": "also-described-by", "target": "family:lie-c", "label": "C₂ low-rank coincidence", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 10, "rank": 2, "cartanType": "B₂ = C₂", "simple": true } }, { "id": "algebra:c-3-c", "slug": "c-3-c", "name": "sp(6, C) — complex Lie algebra", "symbol": "sp(6, C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "C₃", "sp(6, C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex 6 × 6 matrices X satisfying XᵀJ + JX = 0 for a nondegenerate alternating form J.", "definition": "The complex 6 × 6 matrices X satisfying XᵀJ + JX = 0 for a nondegenerate alternating form J.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "21", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "C₃", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-c", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 21, "rank": 3, "cartanType": "C₃", "simple": true } }, { "id": "algebra:d-4-c", "slug": "d-4-c", "name": "so(8, C) — complex Lie algebra", "symbol": "so(8, C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "D₄", "so(8, C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex 8 × 8 matrices skew with respect to a nondegenerate symmetric bilinear form.", "definition": "The complex 8 × 8 matrices skew with respect to a nondegenerate symmetric bilinear form.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "28", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "D₄", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-d", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 28, "rank": 4, "cartanType": "D₄", "simple": true } }, { "id": "algebra:d-5-c", "slug": "d-5-c", "name": "so(10, C) — complex Lie algebra", "symbol": "so(10, C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "D₅", "so(10, C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex 10 × 10 matrices skew with respect to a nondegenerate symmetric bilinear form.", "definition": "The complex 10 × 10 matrices skew with respect to a nondegenerate symmetric bilinear form.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "45", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "5", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "D₅", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-d", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 45, "rank": 5, "cartanType": "D₅", "simple": true } }, { "id": "algebra:e-6-c", "slug": "e-6-c", "name": "e₆(C) — complex Lie algebra", "symbol": "e₆(C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "E₆", "e₆(C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex simple Lie algebra with Dynkin diagram E₆.", "definition": "The complex simple Lie algebra with Dynkin diagram E₆.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "78", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "6", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "E₆", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-e6", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 78, "rank": 6, "cartanType": "E₆", "simple": true } }, { "id": "algebra:e-7-c", "slug": "e-7-c", "name": "e₇(C) — complex Lie algebra", "symbol": "e₇(C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "E₇", "e₇(C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex simple Lie algebra with Dynkin diagram E₇.", "definition": "The complex simple Lie algebra with Dynkin diagram E₇.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "133", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "7", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "E₇", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-e7", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 133, "rank": 7, "cartanType": "E₇", "simple": true } }, { "id": "algebra:e-8-c", "slug": "e-8-c", "name": "e₈(C) — complex Lie algebra", "symbol": "e₈(C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "E₈", "e₈(C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex simple Lie algebra with Dynkin diagram E₈.", "definition": "The complex simple Lie algebra with Dynkin diagram E₈.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "248", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "8", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "E₈", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-e8", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 248, "rank": 8, "cartanType": "E₈", "simple": true } }, { "id": "algebra:f-4-c", "slug": "f-4-c", "name": "f₄(C) — complex Lie algebra", "symbol": "f₄(C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "F₄", "f₄(C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex simple Lie algebra with Dynkin diagram F₄.", "definition": "The complex simple Lie algebra with Dynkin diagram F₄.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "52", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "F₄", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-f4", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 52, "rank": 4, "cartanType": "F₄", "simple": true } }, { "id": "algebra:g-2-c", "slug": "g-2-c", "name": "g₂(C) — complex Lie algebra", "symbol": "g₂(C)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "G₂", "g₂(C)" ], "tags": [ "Complex simple algebra" ], "summary": "The complex simple Lie algebra with Dynkin diagram G₂.", "definition": "The complex simple Lie algebra with Dynkin diagram G₂.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "14", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "G₂", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Choose a real-form record before selecting a real Lie group. Highest weights belong to a weight-lattice convention, and their integration to a global quotient requires a kernel test.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "member-of", "target": "family:lie-g2", "label": "Simple-type classification", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 14, "rank": 2, "cartanType": "G₂", "simple": true } }, { "id": "algebra:a-1-plus-a-1-c", "slug": "a-1-plus-a-1-c", "name": "sl(2, C) ⊕ sl(2, C) — complex Lie algebra", "symbol": "sl(2, C) ⊕ sl(2, C)", "category": "lie", "level": "reference", "aliases": [ "so(4,C)", "D₂" ], "searchTerms": [ "A₁ ⊕ A₁", "sl(2, C) ⊕ sl(2, C)" ], "tags": [ "Reductive algebra" ], "summary": "The direct sum of two complex sl(2,C) ideals; equivalently so(4,C). This algebra is semisimple and not simple.", "definition": "The direct sum of two complex sl(2,C) ideals; equivalently so(4,C). This algebra is semisimple and not simple.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "6", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "A₁ ⊕ A₁", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "No", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Direct sums and central summands are retained explicitly so semisimple and reductive examples do not appear in the simple-family classification.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 6, "rank": 2, "cartanType": "A₁ ⊕ A₁", "simple": false } }, { "id": "algebra:abelian-1-c", "slug": "abelian-1-c", "name": "C (zero bracket) — complex Lie algebra", "symbol": "C (zero bracket)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "Abelian", "C (zero bracket)" ], "tags": [ "Reductive algebra" ], "summary": "The one-dimensional complex vector space with zero Lie bracket. It is reductive and not semisimple.", "definition": "The one-dimensional complex vector space with zero Lie bracket. It is reductive and not semisimple.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "1", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "Abelian", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "No", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Direct sums and central summands are retained explicitly so semisimple and reductive examples do not appear in the simple-family classification.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 1, "rank": 1, "cartanType": "Abelian", "simple": false } }, { "id": "algebra:a-1-plus-abelian-1-c", "slug": "a-1-plus-abelian-1-c", "name": "sl(2, C) ⊕ C — complex Lie algebra", "symbol": "sl(2, C) ⊕ C", "category": "lie", "level": "reference", "aliases": [ "gl(2,C)" ], "searchTerms": [ "A₁ plus center", "sl(2, C) ⊕ C" ], "tags": [ "Reductive algebra" ], "summary": "The reductive complex Lie algebra gl(2,C), decomposed into its traceless ideal and scalar center.", "definition": "The reductive complex Lie algebra gl(2,C), decomposed into its traceless ideal and scalar center.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "C", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Complex dimension", "value": "4", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Complex rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "cartan-type", "label": "Root-system type", "value": "A₁ plus center", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over C", "value": "No", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ], "sections": [ { "id": "interpretation", "title": "Using the algebra record", "paragraphs": [ { "text": "The dimension here is complex dimension. A real form has the same real dimension, while restriction of scalars of this complex algebra to R doubles the dimension. Its connected integrating group and center require additional global information.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Direct sums and central summands are retained explicitly so semisimple and reductive examples do not appear in the simple-family classification.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ] } ], "relationships": [], "capabilities": [], "kind": "algebra", "data": { "field": "C", "dimension": 4, "rank": 2, "cartanType": "A₁ plus center", "simple": false } }, { "id": "algebra:su-2-r", "slug": "su-2-r", "name": "su(2) — real Lie algebra", "symbol": "su(2)", "category": "lie", "level": "reference", "aliases": [ "so(3)", "sp(1)" ], "searchTerms": [], "tags": [ "Compact algebra", "Simple over R" ], "summary": "Traceless skew-Hermitian 2 × 2 complex matrices, regarded as a real Lie algebra.", "definition": "Traceless skew-Hermitian 2 × 2 complex matrices, regarded as a real Lie algebra.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "1", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(2, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "0", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:a-1-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form-record", "target": "real-form:su-2-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:su-2", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:so-3", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 3, "rank": 1, "cartanType": "A₁", "simple": true } }, { "id": "algebra:su-3-r", "slug": "su-3-r", "name": "su(3) — real Lie algebra", "symbol": "su(3)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Compact algebra", "Simple over R" ], "summary": "Traceless skew-Hermitian 3 × 3 complex matrices, regarded as a real Lie algebra.", "definition": "Traceless skew-Hermitian 3 × 3 complex matrices, regarded as a real Lie algebra.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "8", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(3, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "0", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:a-2-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form-record", "target": "real-form:su-3-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:su-3", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:psu-3", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 8, "rank": 2, "cartanType": "A₂", "simple": true } }, { "id": "algebra:su-4-r", "slug": "su-4-r", "name": "su(4) ≅ so(6) — real Lie algebra", "symbol": "su(4) ≅ so(6)", "category": "lie", "level": "reference", "aliases": [ "so(6)" ], "searchTerms": [], "tags": [ "Compact algebra", "Simple over R" ], "summary": "Traceless skew-Hermitian 4 × 4 complex matrices, regarded as a real Lie algebra; isomorphic to the compact orthogonal algebra so(6).", "definition": "Traceless skew-Hermitian 4 × 4 complex matrices, regarded as a real Lie algebra; isomorphic to the compact orthogonal algebra so(6).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(4, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "0", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:a-3-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form-record", "target": "real-form:su-4-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:su-4", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:so-6", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:psu-4", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 15, "rank": 3, "cartanType": "A₃", "simple": true } }, { "id": "algebra:su-5-r", "slug": "su-5-r", "name": "su(5) — real Lie algebra", "symbol": "su(5)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Compact algebra", "Simple over R" ], "summary": "Traceless skew-Hermitian 5 × 5 complex matrices, regarded as a real Lie algebra.", "definition": "Traceless skew-Hermitian 5 × 5 complex matrices, regarded as a real Lie algebra.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "24", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(5, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "0", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:a-4-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form-record", "target": "real-form:su-5-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:su-5", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 24, "rank": 4, "cartanType": "A₄", "simple": true } }, { "id": "algebra:sp-2-compact-r", "slug": "sp-2-compact-r", "name": "sp(2) ≅ so(5) — real Lie algebra", "symbol": "sp(2) ≅ so(5)", "category": "lie", "level": "reference", "aliases": [ "so(5)", "usp(4)" ], "searchTerms": [], "tags": [ "Compact algebra", "Simple over R" ], "summary": "The compact quaternionic skew-Hermitian 2 × 2 matrices; isomorphic to so(5). This is not sp(4,R).", "definition": "The compact quaternionic skew-Hermitian 2 × 2 matrices; isomorphic to so(5). This is not sp(4,R).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "so(5, C) ≅ sp(4, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "0", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. 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Its abstract Lie algebra alone does not distinguish the integrating groups U(1) and R.", "definition": "The real one-dimensional abelian Lie algebra iR with zero bracket. Its abstract Lie algebra alone does not distinguish the integrating groups U(1) and R.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "1", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "C (zero bracket)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. 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They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "For an abelian central summand, the abstract real Lie algebra does not encode whether the corresponding connected group factor is a circle or an additive real line. This record therefore links the chosen compact integration explicitly.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:a-1-plus-abelian-1-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:u-2", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 4, "rank": 2, "cartanType": "A₁ plus center", "simple": false } }, { "id": "algebra:sl-2-r", "slug": "sl-2-r", "name": "sl(2, R) — real Lie algebra", "symbol": "sl(2, R)", "category": "lie", "level": "reference", "aliases": [ "su(1,1)", "so(2,1)" ], "searchTerms": [], "tags": [ "Noncompact algebra", "Simple over R" ], "summary": "The real traceless 2 × 2 matrices with commutator bracket, the split real form of sl(2,C).", "definition": "The real traceless 2 × 2 matrices with commutator bracket, the split real form of sl(2,C).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "1", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(2, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:a-1-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form-record", "target": "real-form:sl-2-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:sl-2-r", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:psl-2-r", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:universal-cover-sl-2-r", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 3, "rank": 1, "cartanType": "A₁", "simple": true } }, { "id": "algebra:sl-3-r", "slug": "sl-3-r", "name": "sl(3, R) — real Lie algebra", "symbol": "sl(3, R)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Noncompact algebra", "Simple over R" ], "summary": "The real traceless 3 × 3 matrices with commutator bracket, the split real form of sl(3,C).", "definition": "The real traceless 3 × 3 matrices with commutator bracket, the split real form of sl(3,C).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "8", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(3, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:a-2-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form-record", "target": "real-form:sl-3-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:sl-3-r", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 8, "rank": 2, "cartanType": "A₂", "simple": true } }, { "id": "algebra:su-2-2-r", "slug": "su-2-2-r", "name": "su(2,2) ≅ so(2,4) — real Lie algebra", "symbol": "su(2,2) ≅ so(2,4)", "category": "lie", "level": "reference", "aliases": [ "so(2,4)", "so(4,2)" ], "searchTerms": [], "tags": [ "Noncompact algebra", "Simple over R" ], "summary": "The real algebra of traceless complex 4 × 4 matrices X satisfying X†H + HX = 0 for H = diag(1,1,−1,−1). It is isomorphic to so(2,4).", "definition": "The real algebra of traceless complex 4 × 4 matrices X satisfying X†H + HX = 0 for H = diag(1,1,−1,−1). It is isomorphic to so(2,4).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(4, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:a-3-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form-record", "target": "real-form:su-2-2-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:su-2-2", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:so-2-4-identity", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 15, "rank": 3, "cartanType": "A₃", "simple": true } }, { "id": "algebra:sp-4-r", "slug": "sp-4-r", "name": "sp(4, R) ≅ so(2,3) — real Lie algebra", "symbol": "sp(4, R) ≅ so(2,3)", "category": "lie", "level": "reference", "aliases": [ "so(2,3)", "so(3,2)" ], "searchTerms": [], "tags": [ "Noncompact algebra", "Simple over R" ], "summary": "The real 4 × 4 matrices X satisfying XᵀJ + JX = 0 for a nondegenerate alternating J. This split B₂ = C₂ algebra is isomorphic to so(2,3).", "definition": "The real 4 × 4 matrices X satisfying XᵀJ + JX = 0 for a nondegenerate alternating J. This split B₂ = C₂ algebra is isomorphic to so(2,3).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "so(5, C) ≅ sp(4, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "complexifies-to", "target": "algebra:b-2-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form-record", "target": "real-form:sp-4-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:sp-4-r", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:so-2-3-identity", "label": "Curated integrating group", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 10, "rank": 2, "cartanType": "B₂ = C₂", "simple": true } }, { "id": "algebra:sl-2-c-as-real", "slug": "sl-2-c-as-real", "name": "sl(2, C) viewed over R — real Lie algebra", "symbol": "sl(2, C) viewed over R", "category": "lie", "level": "reference", "aliases": [ "so(1,3)", "so(3,1)" ], "searchTerms": [], "tags": [ "Noncompact algebra", "Simple over R" ], "summary": "The complex traceless 2 × 2 matrices with only real scalar multiplication retained. This is a simple real algebra, isomorphic to so(1,3); its complexification is sl(2,C) ⊕ sl(2,C).", "definition": "The complex traceless 2 × 2 matrices with only real scalar multiplication retained. This is a simple real algebra, isomorphic to so(1,3); its complexification is sl(2,C) ⊕ sl(2,C).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: complex simple Lie algebras viewed as simple real algebras" } ], "properties": [ { "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "rank", "label": "Absolute rank (of complexification)", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real algebra", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(2, C) ⊕ sl(2, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "integration", "title": "From algebra to group", "paragraphs": [ { "text": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "Restriction of scalars doubles the real dimension. Complexifying this realification gives two complex sl(2,C) factors, which label the two chiral highest weights. It is not a real form of the single complex A₁ algebra.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 9.4: real forms and complexification" }, { "source": "knapp-structure", "locator": "Theorem 6.105: complex simple algebras considered as real algebras" } ], "derivation": { "method": "For a complex algebra g, g_R ⊗_R C ≅ g ⊕ conjugate(g). 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The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "realizes-algebra", "target": "algebra:sl-3-r", "label": "Actual real Lie algebra", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "real-form-of", "target": "algebra:a-2-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:sl-3-r", "label": "Curated global form", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "real-form", "data": { "complexification": "sl(3, C)", "complexificationIds": [ "algebra:a-2-c" ], "compact": false, "realRank": 2, "integrationIds": [ "group:sl-3-r" ], "realAlgebraId": "algebra:sl-3-r" } }, { "id": "real-form:su-2-2-r", "slug": "su-2-2-r", "name": "su(2,2) ≅ so(2,4) real form", "symbol": "su(2,2) ≅ so(2,4)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Noncompact real form" ], "summary": "Noncompact real form of sl(4, C) with its curated integrating groups.", "definition": "The real algebra of traceless complex 4 × 4 matrices X satisfying X†H + HX = 0 for H = diag(1,1,−1,−1). It is isomorphic to so(2,4).", "definitionEvidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(4, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real form", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "global-forms", "title": "One algebra, several integrations", "paragraphs": [ { "text": "A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "realizes-algebra", "target": "algebra:su-2-2-r", "label": "Actual real Lie algebra", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "real-form-of", "target": "algebra:a-3-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:su-2-2", "label": "Curated global form", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:so-2-4-identity", "label": "Curated global form", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "real-form", "data": { "complexification": "sl(4, C)", "complexificationIds": [ "algebra:a-3-c" ], "compact": false, "realRank": 2, "integrationIds": [ "group:su-2-2", "group:so-2-4-identity" ], "realAlgebraId": "algebra:su-2-2-r" } }, { "id": "real-form:sp-4-r", "slug": "sp-4-r", "name": "sp(4, R) ≅ so(2,3) real form", "symbol": "sp(4, R) ≅ so(2,3)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Noncompact real form" ], "summary": "Noncompact real form of so(5, C) ≅ sp(4, C) with its curated integrating groups.", "definition": "The real 4 × 4 matrices X satisfying XᵀJ + JX = 0 for a nondegenerate alternating J. This split B₂ = C₂ algebra is isomorphic to so(2,3).", "definitionEvidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "so(5, C) ≅ sp(4, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real form", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "global-forms", "title": "One algebra, several integrations", "paragraphs": [ { "text": "A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "realizes-algebra", "target": "algebra:sp-4-r", "label": "Actual real Lie algebra", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "real-form-of", "target": "algebra:b-2-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:sp-4-r", "label": "Curated global form", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:so-2-3-identity", "label": "Curated global form", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "real-form", "data": { "complexification": "so(5, C) ≅ sp(4, C)", "complexificationIds": [ "algebra:b-2-c" ], "compact": false, "realRank": 2, "integrationIds": [ "group:sp-4-r", "group:so-2-3-identity" ], "realAlgebraId": "algebra:sp-4-r" } }, { "id": "real-form:sl-2-c-as-real", "slug": "sl-2-c-as-real", "name": "sl(2, C) viewed over R real form", "symbol": "sl(2, C) viewed over R", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Noncompact real form" ], "summary": "Noncompact real form of sl(2, C) ⊕ sl(2, C) with its curated integrating groups.", "definition": "The complex traceless 2 × 2 matrices with only real scalar multiplication retained. This is a simple real algebra, isomorphic to so(1,3); its complexification is sl(2,C) ⊕ sl(2,C).", "definitionEvidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: complex simple Lie algebras viewed over R" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "complexification", "label": "Complexification", "value": "sl(2, C) ⊕ sl(2, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "compact", "label": "Compact real form", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "global-forms", "title": "One algebra, several integrations", "paragraphs": [ { "text": "A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "text": "Here the complex algebra is A₁ ⊕ A₁. The same complexification also has the compact real form su(2) ⊕ su(2), but the two real algebras are different: the Lorentz algebra is simple over R and the compact one is a direct sum.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105: real simple classification" } ], "derivation": { "method": "Compare the ideals of the realification with those of the compact direct sum, and complexify both.", "inputs": [ "algebra:sl-2-c-as-real", "algebra:so-4-compact-r", "algebra:a-1-plus-a-1-c" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "realizes-algebra", "target": "algebra:sl-2-c-as-real", "label": "Actual real Lie algebra", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "real-form-of", "target": "algebra:a-1-plus-a-1-c", "label": "Complexification", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "integrates-to", "target": "group:sl-2-c", "label": "Curated global form", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "integrates-to", "target": "group:so-1-3-identity", "label": "Curated global form", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "real-form", "data": { "complexification": "sl(2, C) ⊕ sl(2, C)", "complexificationIds": [ "algebra:a-1-plus-a-1-c" ], "compact": false, "realRank": 1, "integrationIds": [ "group:sl-2-c", "group:so-1-3-identity" ], "realAlgebraId": "algebra:sl-2-c-as-real" } }, { "id": "group:su-4", "slug": "su-4", "name": "Special unitary group in four dimensions", "symbol": "SU(4)", "category": "lie", "level": "executable", "aliases": [ "Spin(6)" ], "searchTerms": [ "su(4) ≅ so(6)" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected compact form of A₃. Complex unitary 4 × 4 matrices with determinant 1.", "definition": "Complex unitary 4 × 4 matrices with determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₄", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected compact form of A₃", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(4) ≅ so(6)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "SU(4) is simply connected. Its center consists of the scalar fourth roots of unity. The identification Spin(6) ≅ SU(4) gives the vector cover SU(4) → SO(6) with kernel {±I}.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Finite-dimensional complex irreducibles have A₃ dominant integral highest weights. The defining 4 and its dual are the two chiral spin representations in the Spin(6) identification; the exterior square of the 4 is the vector 6.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "garrett-isogenies", "locator": "Section 2.9: SU(4) acting on the real form of its exterior square" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:su-4-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:su-4-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:su-4-to-so-6", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "has-map", "target": "map:su-4-to-psu-4", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [ "lie-representations", "matter-quotients", "sun-tensor-products", "charged-branching" ], "kind": "group", "data": { "category": "lie", "dimension": 15, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:su-4-r", "capabilityPresets": { "lie-representations": { "type": "A", "rank": "3", "labels": "1,0,0" }, "matter-quotients": { "n": "4", "matter": "1,0,1", "quotient": "1" }, "sun-tensor-products": { "n": "4", "left": "1,0,0", "right": "0,0,1" }, "charged-branching": { "n": "4", "p": "3", "labels": "1,0,0" } } } }, { "id": "group:psu-4", "slug": "psu-4", "name": "Adjoint compact group of type A₃", "symbol": "PSU(4)", "category": "lie", "level": "reference", "aliases": [ "SU(4)/C4", "PSO(6)" ], "searchTerms": [ "su(4) ≅ so(6)" ], "tags": [ "Compact Lie group", "Simple algebra", "Global form" ], "summary": "SU(4)/C₄. The quotient of SU(4) by its full scalar center C₄.", "definition": "The quotient of SU(4) by its full scalar center C₄.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₄", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "SU(4)/C₄", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(4) ≅ so(6)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Its simply connected cover is SU(4), with covering kernel C₄. Through Spin(6) ≅ SU(4), this quotient is PSO(6) = SO(6)/{±I}.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "An SU(4) highest weight with Dynkin labels (a,b,c) descends exactly when a + 2b + 3c is divisible by 4. This criterion concerns the center action, not the reality type.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "derivation": { "method": "The k-th exterior-power fundamental representation has scalar-center charge k; add charges with the Dynkin-label multiplicities.", "inputs": [ "The center generator iI of SU(4)", "Fundamental weights realized by Λ¹, Λ², and Λ³ of C⁴" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:su-4-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:su-4-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:su-4-to-psu-4", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "covered-by", "target": "group:su-4", "label": "Specified connected covering group", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 15, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:su-4-r" } }, { "id": "group:su-5", "slug": "su-5", "name": "Special unitary group in five dimensions", "symbol": "SU(5)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "su(5)" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected compact form of A₄. Complex unitary 5 × 5 matrices with determinant 1.", "definition": "Complex unitary 5 × 5 matrices with determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "24", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₅", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected compact form of A₄", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(5)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "SU(5) is simply connected with scalar center generated by exp(2πi/5)I. Its adjoint global form is the quotient by this center.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Dominant integral A₄ highest weights label its irreducibles. For Dynkin labels (a₁,a₂,a₃,a₄), the scalar-center charge is a₁ + 2a₂ + 3a₃ + 4a₄ modulo 5.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "derivation": { "method": "The k-th exterior power of the defining representation has center charge k; charges add under highest-weight addition.", "inputs": [ "Scalar center generator exp(2πi/5)I", "Fundamental representations ΛᵏC⁵, k=1,2,3,4" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:su-5-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:su-5-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [ "lie-representations", "matter-quotients", "sun-tensor-products", "charged-branching" ], "kind": "group", "data": { "category": "lie", "dimension": 24, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:su-5-r", "capabilityPresets": { "lie-representations": { "type": "A", "rank": "4", "labels": "1,0,0,0" }, "matter-quotients": { "n": "5", "matter": "1,0,0,1", "quotient": "1" }, "sun-tensor-products": { "n": "5", "left": "1,0,0,0", "right": "0,0,0,1" }, "charged-branching": { "n": "5", "p": "3", "labels": "1,0,0,0" } } } }, { "id": "group:u-2", "slug": "u-2", "name": "Unitary group in two dimensions", "symbol": "U(2)", "category": "lie", "level": "reference", "aliases": [ "Unitary 2 × 2 group" ], "searchTerms": [ "u(2)" ], "tags": [ "Compact Lie group", "Reductive algebra", "Global form" ], "summary": "(SU(2) × U(1))/diagonal C₂. All complex unitary 2 × 2 matrices, with no determinant restriction.", "definition": "All complex unitary 2 × 2 matrices, with no determinant restriction.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "4", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "U(1)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "(SU(2) × U(1))/diagonal C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "u(2)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The surjective map (A,z) ↦ zA from SU(2) × U(1) has kernel {(I,1),(−I,−1)}. The determinant map U(2) → U(1) has fiber SU(2), and induces an isomorphism on fundamental groups.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Use the determinant fibration with simply connected fiber SU(2), or its section diag(z,1); apply the homotopy exact sequence.", "inputs": [ "group:su-2#pi1", "group:u-1#pi1", "Determinant map U(2) → U(1)" ] }, "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Pulling an irrep back to SU(2) × U(1) gives a highest weight n ≥ 0 and an integer U(1) charge q with n + q even. The parity condition makes (−I,−1) act trivially. Its complex dimension is n + 1.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "derivation": { "method": "The diagonal kernel acts by (−1)ⁿ(−1)ᑫ, so descent requires n+q even.", "inputs": [ "SU(2) highest weight n has dimension n+1 and central sign (−1)ⁿ", "U(1) character z ↦ zᑫ", "U(2) quotient kernel {(I,1),(−I,−1)}" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:u-2-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 4, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:u-2-r" } }, { "id": "group:so-4", "slug": "so-4", "name": "Rotation group in four dimensions", "symbol": "SO(4)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "su(2) ⊕ su(2)" ], "tags": [ "Compact Lie group", "Reductive algebra", "Global form" ], "summary": "(SU(2) × SU(2))/diagonal C₂. The real orthogonal 4 × 4 matrices of determinant 1.", "definition": "The real orthogonal 4 × 4 matrices of determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "(SU(2) × SU(2))/diagonal C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic construction of Spin(4) → SO(4)" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(2) ⊕ su(2)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The universal cover Spin(4) = SU(2) × SU(2) acts on quaternions by x ↦ axb⁻¹. Its kernel is {(1,1),(−1,−1)}. The center of SO(4) is {±I}.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic construction of Spin(4) → SO(4)" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "An irreducible Spin(4) representation (jL,jR) descends to SO(4) exactly when jL + jR is an integer. The dimension is (2jL + 1)(2jR + 1).", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:so-4-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:so-4-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:spin-4-to-so-4", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic construction of Spin(4) → SO(4)" } ] }, { "type": "covered-by", "target": "group:spin-4", "label": "Specified connected covering group", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic construction of Spin(4) → SO(4)" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 6, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:so-4-compact-r" } }, { "id": "group:so-5", "slug": "so-5", "name": "Rotation group in five dimensions", "symbol": "SO(5)", "category": "lie", "level": "reference", "aliases": [ "PSp(2)" ], "searchTerms": [ "sp(2) ≅ so(5)" ], "tags": [ "Compact Lie group", "Simple algebra", "Global form" ], "summary": "Sp(2)/{±I}. The real orthogonal 5 × 5 matrices of determinant 1.", "definition": "The real orthogonal 5 × 5 matrices of determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Sp(2)/{±I}", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.6: compact Sp(2) → SO(5)" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sp(2) ≅ so(5)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The exceptional isomorphism Spin(5) ≅ Sp(2) realizes the universal double cover. The compact symplectic group in this statement acts on quaternionic dimension two.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.6: compact Sp(2) → SO(5)" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Use finite-dimensional continuous complex representations. An irreducible representation of the simply connected compact cover descends precisely when the specified covering kernel acts trivially; the same Lie algebra does not remove that global condition.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sp-2-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sp-2-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:sp-2-to-so-5", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.6: compact Sp(2) → SO(5)" } ] }, { "type": "covered-by", "target": "group:sp-2", "label": "Specified connected covering group", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.6: compact Sp(2) → SO(5)" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 10, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:sp-2-compact-r" } }, { "id": "group:so-6", "slug": "so-6", "name": "Rotation group in six dimensions", "symbol": "SO(6)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "su(4) ≅ so(6)" ], "tags": [ "Compact Lie group", "Simple algebra", "Global form" ], "summary": "SU(4)/{±I}. The real orthogonal 6 × 6 matrices of determinant 1.", "definition": "The real orthogonal 6 × 6 matrices of determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "SU(4)/{±I}", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(4) ≅ so(6)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "SU(4) ≅ Spin(6) is its universal double cover. Dividing further by the remaining order-two center gives PSO(6) ≅ PSU(4); this is a different global form.", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Use finite-dimensional continuous complex representations. An irreducible representation of the simply connected compact cover descends precisely when the specified covering kernel acts trivially; the same Lie algebra does not remove that global condition.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:su-4-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:su-4-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:su-4-to-so-6", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] }, { "type": "covered-by", "target": "group:su-4", "label": "Specified connected covering group", "evidence": [ { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 15, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:su-4-r" } }, { "id": "group:sp-2", "slug": "sp-2", "name": "Compact symplectic group of quaternionic rank two", "symbol": "Sp(2)", "category": "lie", "level": "executable", "aliases": [ "USp(4)", "Spin(5)" ], "searchTerms": [ "sp(2) ≅ so(5)" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected compact form of B₂ = C₂. Quaternionic unitary 2 × 2 matrices. Equivalently, this is the compact subgroup of U(4) preserving a complex symplectic form.", "definition": "Quaternionic unitary 2 × 2 matrices. Equivalently, this is the compact subgroup of U(4) preserving a complex symplectic form.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected compact form of B₂ = C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sp(2) ≅ so(5)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Sp(2) is compact and simply connected, with center {±I}. The isomorphism Spin(5) ≅ Sp(2) is an isomorphism of global groups, not only of Lie algebras.", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The defining complex 4 preserves a nondegenerate antisymmetric form and has quaternionic type. The vector 5 is the traceless part of the exterior square of this 4; it factors through SO(5).", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "garrett-isogenies", "locator": "Section 2.6: five-dimensional orthogonal action of compact Sp(2)" } ], "assumptions": [] } ] }, { "id": "notation", "title": "Symplectic notation", "paragraphs": [ { "text": "This site writes compact Sp(n) for quaternionic n × n unitary matrices, also called USp(2n). It writes Sp(2n,R) for real matrices of size 2n preserving an alternating form. The two conventions count different dimensions.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sp-2-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sp-2-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:sp-2-to-so-5", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" } ] } ], "capabilities": [ "lie-representations" ], "kind": "group", "data": { "category": "lie", "dimension": 10, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:sp-2-compact-r", "capabilityPresets": { "lie-representations": { "type": "C", "rank": "2", "labels": "1,0" } } } }, { "id": "group:sp-3", "slug": "sp-3", "name": "Compact symplectic group of quaternionic rank three", "symbol": "Sp(3)", "category": "lie", "level": "reference", "aliases": [ "USp(6)" ], "searchTerms": [ "sp(3)" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected compact form of C₃. Quaternionic unitary 3 × 3 matrices; equivalently the compact unitary symplectic subgroup of U(6).", "definition": "Quaternionic unitary 3 × 3 matrices; equivalently the compact unitary symplectic subgroup of U(6).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "21", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected compact form of C₃", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sp(3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Sp(3) is compact and simply connected with center {±I}. In this notation the argument 3 is quaternionic dimension, while its defining complex representation has dimension 6.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The defining complex 6 has quaternionic type. All irreducible complex representations are indexed by dominant integral C₃ highest weights; a central quotient keeps those for which −I acts trivially.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] }, { "id": "notation", "title": "Symplectic notation", "paragraphs": [ { "text": "This site writes compact Sp(n) for quaternionic n × n unitary matrices, also called USp(2n). It writes Sp(2n,R) for real matrices of size 2n preserving an alternating form. The two conventions count different dimensions.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sp-3-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sp-3-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 21, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:sp-3-compact-r" } }, { "id": "group:spin-8", "slug": "spin-8", "name": "Simply connected compact spin group in eight dimensions", "symbol": "Spin(8)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "so(8)" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected compact form of D₄. The compact spin group in the even Clifford algebra of Euclidean R⁸, with its standard vector action on R⁸.", "definition": "The compact spin group in the even Clifford algebra of Euclidean R⁸, with its standard vector action on R⁸.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "28", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂ × C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected compact form of D₄", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "so(8)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The vector homomorphism Spin(8) → SO(8) has kernel {±1}. Its full center is C₂ × C₂, and quotienting by that full center gives PSO(8).", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The vector representation and the two half-spin representations each have dimension 8. Triality permutes these representations and their order-two central kernels. Therefore the three intermediate central quotients are isomorphic as abstract groups, while their specified covering maps carry different representation information.", "evidence": [ { "source": "conrad", "locator": "Appendix P: spin representations; Remark T.2.1, printed page 262: D₄ triality and central kernels" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:so-8-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:so-8-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:spin-8-to-so-8", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [ "lie-representations" ], "kind": "group", "data": { "category": "lie", "dimension": 28, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:so-8-compact-r", "capabilityPresets": { "lie-representations": { "type": "D", "rank": "4", "labels": "1,0,0,0" } } } }, { "id": "group:so-8", "slug": "so-8", "name": "Rotation group in eight dimensions", "symbol": "SO(8)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "so(8)" ], "tags": [ "Compact Lie group", "Simple algebra", "Global form" ], "summary": "Spin(8)/{±1} for the vector-cover kernel. The real orthogonal 8 × 8 matrices of determinant 1.", "definition": "The real orthogonal 8 × 8 matrices of determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "28", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Spin(8)/{±1} for the vector-cover kernel", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "so(8)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Its universal cover is Spin(8) with the vector-cover kernel {±1}. The remaining center {±I} gives the quotient PSO(8). Triality identifies other order-two Spin(8) quotients abstractly, without identifying their chosen vector or spinor actions.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Use finite-dimensional continuous complex representations. An irreducible representation of the simply connected compact cover descends precisely when the specified covering kernel acts trivially; the same Lie algebra does not remove that global condition.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:so-8-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:so-8-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:spin-8-to-so-8", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality" } ] }, { "type": "covered-by", "target": "group:spin-8", "label": "Specified connected covering group", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 28, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:so-8-compact-r" } }, { "id": "group:pso-8", "slug": "pso-8", "name": "Adjoint compact group of type D₄", "symbol": "PSO(8)", "category": "lie", "level": "reference", "aliases": [ "SO(8)/{±I}" ], "searchTerms": [ "so(8)" ], "tags": [ "Compact Lie group", "Simple algebra", "Global form" ], "summary": "Spin(8)/(C₂ × C₂). The quotient of SO(8) by its center {±I}, equivalently the adjoint quotient of Spin(8).", "definition": "The quotient of SO(8) by its center {±I}, equivalently the adjoint quotient of Spin(8).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "28", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂ × C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Spin(8)/(C₂ × C₂)", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "so(8)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The simply connected cover is Spin(8), and the covering kernel is its full center C₂ × C₂. Thus the fundamental group has four elements and is not cyclic.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "A Spin(8) irrep descends to PSO(8) precisely when its highest weight lies in the D₄ root lattice. In particular, the vector and the two half-spin 8s do not descend, whereas the adjoint 28 does.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: D₄ center, central kernels, and triality" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:so-8-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:so-8-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "covered-by", "target": "group:spin-8", "label": "Specified connected covering group", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 28, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:so-8-compact-r" } }, { "id": "group:spin-10", "slug": "spin-10", "name": "Simply connected compact spin group in ten dimensions", "symbol": "Spin(10)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "so(10)" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected compact form of D₅. The compact spin group in the even Clifford algebra of Euclidean R¹⁰, with its standard vector action.", "definition": "The compact spin group in the even Clifford algebra of Euclidean R¹⁰, with its standard vector action.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "45", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "5", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₄", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected compact form of D₅", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "so(10)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Spin(10) is simply connected and its center is cyclic of order 4. The element −1 is the order-two kernel of the vector cover to SO(10); quotienting by it leaves an order-two center.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The vector representation has dimension 10 and factors through SO(10). The two complex half-spin representations have dimension 16 and do not factor through SO(10), because the covering element −1 acts as minus the identity on spinors.", "evidence": [ { "source": "conrad", "locator": "Appendix P: Clifford modules and half-spin representations; dim S± = 2^(r−1) in dimension 2r" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:so-10-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:so-10-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:spin-10-to-so-10", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ] } ], "capabilities": [ "lie-representations" ], "kind": "group", "data": { "category": "lie", "dimension": 45, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:so-10-compact-r", "capabilityPresets": { "lie-representations": { "type": "D", "rank": "5", "labels": "1,0,0,0,0" } } } }, { "id": "group:so-10", "slug": "so-10", "name": "Rotation group in ten dimensions", "symbol": "SO(10)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "so(10)" ], "tags": [ "Compact Lie group", "Simple algebra", "Global form" ], "summary": "Spin(10)/{±1}. The real orthogonal 10 × 10 matrices of determinant 1.", "definition": "The real orthogonal 10 × 10 matrices of determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "45", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "5", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Spin(10)/{±1}", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "so(10)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "SO(10) has universal cover Spin(10) and fundamental group C₂. Its matrix center is {±I}. A statement about a spinor 16 must therefore specify the Spin(10) cover.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Use finite-dimensional continuous complex representations. An irreducible representation of the simply connected compact cover descends precisely when the specified covering kernel acts trivially; the same Lie algebra does not remove that global condition.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:so-10-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:so-10-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:spin-10-to-so-10", "label": "Explicit homomorphism", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ] }, { "type": "covered-by", "target": "group:spin-10", "label": "Specified connected covering group", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 45, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:so-10-compact-r" } }, { "id": "group:g-2-compact", "slug": "g-2-compact", "name": "Compact group of octonion automorphisms", "symbol": "G₂ compact", "category": "lie", "level": "executable", "aliases": [ "Aut(O)", "Compact G2" ], "searchTerms": [ "g₂ compact" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected and adjoint compact form of G₂. The automorphism group of the real octonion division algebra O.", "definition": "The automorphism group of the real octonion division algebra O.", "definitionEvidence": [ { "source": "baez-octonions", "locator": "Section 4.1: G₂ as octonion automorphisms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "14", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected and adjoint compact form of G₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "g₂ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The compact G₂ group is simply connected and has trivial center, so its simply connected and adjoint forms coincide.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Octonion automorphisms fix the real unit and preserve the imaginary octonions. Their action on Im(O) is a faithful real representation of dimension 7, preserving its Euclidean norm and octonion multiplication.", "evidence": [ { "source": "baez-octonions", "locator": "Section 4.1: action on imaginary octonions and the subgroup G₂ ⊂ SO(7)" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:g-2-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:g-2-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [ "lie-representations" ], "kind": "group", "data": { "category": "lie", "dimension": 14, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:g-2-compact-r", "capabilityPresets": { "lie-representations": { "type": "G", "rank": "2", "labels": "1,0" } } } }, { "id": "group:f-4-compact", "slug": "f-4-compact", "name": "Compact exceptional group of type F₄", "symbol": "F₄ compact", "category": "lie", "level": "executable", "aliases": [ "Compact F4" ], "searchTerms": [ "f₄ compact" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected and adjoint compact form of F₄. The automorphism group of the Euclidean exceptional Jordan algebra of Hermitian 3 × 3 octonionic matrices, with product X ∘ Y = (XY + YX)/2.", "definition": "The automorphism group of the Euclidean exceptional Jordan algebra of Hermitian 3 × 3 octonionic matrices, with product X ∘ Y = (XY + YX)/2.", "definitionEvidence": [ { "source": "baez-octonions", "locator": "Section 4.2: F₄ and the exceptional Jordan algebra" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "52", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected and adjoint compact form of F₄", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "f₄ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The compact F₄ group is simply connected with trivial center. Its compact global form is therefore unique for this Lie algebra.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The action on the exceptional Jordan algebra fixes its identity. The traceless part gives a real representation of dimension 26.", "evidence": [ { "source": "baez-octonions", "locator": "Section 4.2: the 27-dimensional exceptional Jordan algebra and F₄ automorphisms" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:f-4-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:f-4-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [ "lie-representations" ], "kind": "group", "data": { "category": "lie", "dimension": 52, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:f-4-compact-r", "capabilityPresets": { "lie-representations": { "type": "F", "rank": "4", "labels": "0,0,0,1" } } } }, { "id": "group:e-6-compact-sc", "slug": "e-6-compact-sc", "name": "Simply connected compact exceptional group of type E₆", "symbol": "E₆ compact, simply connected", "category": "lie", "level": "executable", "aliases": [ "Compact simply connected E6" ], "searchTerms": [ "e₆ compact" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected compact form of E₆. The connected simply connected compact Lie group integrating the compact real algebra of type E₆.", "definition": "The connected simply connected compact Lie group integrating the compact real algebra of type E₆.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "78", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "6", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₃", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected compact form of E₆", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₆ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The center is C₃. Its only nontrivial connected central quotient is the adjoint compact E₆/C₃, whose fundamental group is C₃.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Dominant integral E₆ highest weights classify the finite-dimensional complex irreducibles. Passing to the adjoint form retains precisely the weights in the root lattice.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:e-6-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:e-6-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [ "lie-representations" ], "kind": "group", "data": { "category": "lie", "dimension": 78, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:e-6-compact-r", "capabilityPresets": { "lie-representations": { "type": "E", "rank": "6", "labels": "1,0,0,0,0,0" } } } }, { "id": "group:e-7-compact-sc", "slug": "e-7-compact-sc", "name": "Simply connected compact exceptional group of type E₇", "symbol": "E₇ compact, simply connected", "category": "lie", "level": "executable", "aliases": [ "Compact simply connected E7" ], "searchTerms": [ "e₇ compact" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected compact form of E₇. The connected simply connected compact Lie group integrating the compact real algebra of type E₇.", "definition": "The connected simply connected compact Lie group integrating the compact real algebra of type E₇.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "133", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "7", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected compact form of E₇", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₇ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The center is C₂. Quotienting by it gives the adjoint compact E₇/C₂, with fundamental group C₂.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Dominant integral E₇ highest weights classify finite-dimensional complex irreducibles. An irrep descends to the adjoint group exactly when its central sign is positive, equivalently when its highest weight lies in the root lattice.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:e-7-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:e-7-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [ "lie-representations" ], "kind": "group", "data": { "category": "lie", "dimension": 133, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:e-7-compact-r", "capabilityPresets": { "lie-representations": { "type": "E", "rank": "7", "labels": "0,0,0,0,0,0,1" } } } }, { "id": "group:e-8-compact", "slug": "e-8-compact", "name": "Compact exceptional group of type E₈", "symbol": "E₈ compact", "category": "lie", "level": "executable", "aliases": [ "Compact E8" ], "searchTerms": [ "e₈ compact" ], "tags": [ "Compact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected and adjoint compact form of E₈. The connected compact Lie group with compact real Lie algebra of type E₈; its simply connected and adjoint forms agree.", "definition": "The connected compact Lie group with compact real Lie algebra of type E₈; its simply connected and adjoint forms agree.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "248", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "8", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected and adjoint compact form of E₈", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₈ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Its center and fundamental group are both trivial. There is no nontrivial connected central quotient of the simply connected compact E₈ group.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The E₈ weight lattice equals its root lattice. Thus every dominant integral highest weight integrates to this single compact global form; its adjoint representation has dimension 248.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:e-8-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:e-8-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [ "lie-representations" ], "kind": "group", "data": { "category": "lie", "dimension": 248, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:e-8-compact-r", "capabilityPresets": { "lie-representations": { "type": "E", "rank": "8", "labels": "0,0,0,0,0,0,0,1" } } } }, { "id": "group:e-6-compact-ad", "slug": "e-6-compact-ad", "name": "Adjoint compact exceptional group of type E₆", "symbol": "E₆ compact, adjoint", "category": "lie", "level": "reference", "aliases": [ "E6/C3", "E6/Z3" ], "searchTerms": [ "e₆ compact" ], "tags": [ "Compact Lie group", "Simple algebra", "Global form" ], "summary": "E₆ simply connected / C₃. The quotient of the simply connected compact E₆ group by its full center C₃.", "definition": "The quotient of the simply connected compact E₆ group by its full center C₃.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "78", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "6", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₃", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "E₆ simply connected / C₃", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₆ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The adjoint group has trivial center and fundamental group C₃. The cover has the same local real algebra and dimension.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Finite-dimensional complex irreducibles are the dominant E₆ weights lying in the root lattice. This is the zero-center-character sector of the simply connected group.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:e-6-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:e-6-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "covered-by", "target": "group:e-6-compact-sc", "label": "Specified connected covering group", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 78, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:e-6-compact-r" } }, { "id": "group:e-7-compact-ad", "slug": "e-7-compact-ad", "name": "Adjoint compact exceptional group of type E₇", "symbol": "E₇ compact, adjoint", "category": "lie", "level": "reference", "aliases": [ "E7/C2", "E7/Z2" ], "searchTerms": [ "e₇ compact" ], "tags": [ "Compact Lie group", "Simple algebra", "Global form" ], "summary": "E₇ simply connected / C₂. The quotient of the simply connected compact E₇ group by its full center C₂.", "definition": "The quotient of the simply connected compact E₇ group by its full center C₂.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "133", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "rank", "label": "Compact rank", "value": "7", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "E₇ simply connected / C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₇ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The adjoint group has trivial center and fundamental group C₂. This quotient changes global representation descent while preserving the local Lie algebra.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Finite-dimensional complex irreducibles are the dominant E₇ weights lying in the root lattice, equivalently those on which the nonidentity central element of the cover acts trivially.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:e-7-compact-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:e-7-compact-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "covered-by", "target": "group:e-7-compact-sc", "label": "Specified connected covering group", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 133, "compact": true, "connected": true, "abelian": false, "algebraId": "algebra:e-7-compact-r" } }, { "id": "group:sl-2-r", "slug": "sl-2-r", "name": "Real special linear group in two dimensions", "symbol": "SL(2, R)", "category": "lie", "level": "reference", "aliases": [ "SU(1,1)" ], "searchTerms": [ "sl(2, R)", "real rank 1" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Global form" ], "summary": "Matrix form, a quotient of the universal cover by 2Z. The real 2 × 2 matrices of determinant 1.", "definition": "The real 2 × 2 matrices of determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(2) ≅ U(1)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Matrix form, a quotient of the universal cover by 2Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(2, R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(2) ≅ U(1)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Polar decomposition gives the manifold SO(2) × R², so the fundamental group is Z. The center is {±I}. The isomorphism with SU(1,1) is an isomorphism of real matrix Lie groups after a complex change of basis.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Finite-dimensional complex irreducible representations are restrictions of sl(2,C) highest-weight modules with integer highest weight n ≥ 0 and dimension n + 1. The central element −I acts by (−1)ⁿ. These are distinct from infinite-dimensional unitary representations.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-lie-notes", "locator": "Remark 36.7, printed page 196: finite-dimensional representations of the universal cover factor through SL(2,R)" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sl-2-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sl-2-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:sl-2-r-to-psl-2-r", "label": "Explicit homomorphism", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:universal-sl-2-r-to-sl-2-r", "label": "Explicit homomorphism", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "covered-by", "target": "group:universal-cover-sl-2-r", "label": "Specified connected covering group", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 3, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:sl-2-r" } }, { "id": "group:psl-2-r", "slug": "psl-2-r", "name": "Adjoint real group of type A₁", "symbol": "PSL(2, R)", "category": "lie", "level": "reference", "aliases": [ "SO₀(2,1)", "SO₀(1,2)", "PSU(1,1)" ], "searchTerms": [ "sl(2, R)", "real rank 1" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Global form" ], "summary": "SL(2,R)/{±I}. The quotient of SL(2,R) by its scalar center {±I}.", "definition": "The quotient of SL(2,R) by its scalar center {±I}.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(2), with the adjoint angular period", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "SL(2,R)/{±I}", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.2: the SL(2,R) orthogonal action of signature (2,1)" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(2, R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(2), with the adjoint angular period", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Its maximal compact subgroup is again a circle, so it has fundamental group Z. The adjoint action identifies it with SO₀(2,1), the identity component; the notation does not include the other components of O(2,1).", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.2: the SL(2,R) orthogonal action of signature (2,1)" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "An SL(2,R) finite-dimensional complex highest-weight module descends precisely for even highest weight n, so its dimension n + 1 is odd. The complete unitary dual is a different reference problem.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "etingof-lie-notes", "locator": "Remark 36.7: finite-dimensional representations and the universal cover" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sl-2-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sl-2-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:sl-2-r-to-psl-2-r", "label": "Explicit homomorphism", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.2: the SL(2,R) orthogonal action of signature (2,1)" } ] }, { "type": "covered-by", "target": "group:universal-cover-sl-2-r", "label": "Specified connected covering group", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.2: the SL(2,R) orthogonal action of signature (2,1)" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 3, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:sl-2-r" } }, { "id": "group:universal-cover-sl-2-r", "slug": "universal-cover-sl-2-r", "name": "Simply connected group integrating sl(2,R)", "symbol": "Universal cover of SL(2, R)", "category": "lie", "level": "reference", "aliases": [ "Universal cover of PSL(2,R)", "Universal cover of SU(1,1)" ], "searchTerms": [ "sl(2, R)", "real rank 1" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Simply connected" ], "summary": "Simply connected real Lie group; not a matrix group. The simply connected covering Lie group of SL(2,R), or equivalently of PSL(2,R).", "definition": "The simply connected covering Lie group of SL(2,R), or equivalently of PSL(2,R).", "definitionEvidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: the universal covering group of SL(2,R)" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "Z", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: infinite cyclic center and absence of nontrivial compact subgroups" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: universal covering Lie groups" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Simply connected real Lie group; not a matrix group", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: structure of the universal cover" }, { "source": "knapp-structure", "locator": "Theorem 6.46: Iwasawa decomposition" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(2, R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "Trivial", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: infinite cyclic center and absence of nontrivial compact subgroups" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The lifted Iwasawa factors identify its manifold with R³. Let z be the central element represented by a lift of a rotation through π in SL(2,R). Then the projection to SL(2,R) has kernel ⟨z²⟩, while the projection to PSL(2,R) has kernel ⟨z⟩.", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: structure of the universal cover" }, { "source": "knapp-structure", "locator": "Theorem 6.46: Iwasawa decomposition" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Every finite-dimensional complex representation factors through SL(2,R). Consequently this simply connected Lie group has no faithful finite-dimensional matrix representation; its infinite cyclic center is not detected faithfully in that category.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Remark 36.7, printed page 196; Exercise 11.20: finite-dimensional representations and nonlinearity of the universal cover" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sl-2-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sl-2-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:universal-sl-2-r-to-sl-2-r", "label": "Explicit homomorphism", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: structure of the universal cover" }, { "source": "knapp-structure", "locator": "Theorem 6.46: Iwasawa decomposition" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 3, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:sl-2-r" } }, { "id": "group:sl-3-r", "slug": "sl-3-r", "name": "Real special linear group in three dimensions", "symbol": "SL(3, R)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "sl(3, R)", "real rank 2" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Global form" ], "summary": "Split real matrix form of A₂. The real 3 × 3 matrices of determinant 1.", "definition": "The real 3 × 3 matrices of determinant 1.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "8", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(3)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Split real matrix form of A₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(3, R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Polar decomposition identifies the manifold with SO(3) × R⁵. Therefore π₁ is C₂ and this centerless matrix group is not simply connected. Real scalar matrices of determinant 1 give only the identity in its center.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The defining real 3 gives a faithful matrix representation. Finite-dimensional complex irreducibles are restrictions of sl(3,C) highest-weight modules; representation data for infinite-dimensional unitary families are outside this record.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-lie-notes", "locator": "Section 9: Lie-group and Lie-algebra representations; connected integration" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sl-3-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sl-3-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 8, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:sl-3-r" } }, { "id": "group:su-2-2", "slug": "su-2-2", "name": "Indefinite special unitary group of signature (2,2)", "symbol": "SU(2,2)", "category": "lie", "level": "reference", "aliases": [ "Spin₀(2,4)", "Spin₀(4,2)" ], "searchTerms": [ "su(2,2) ≅ so(2,4)", "real rank 2" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Global form" ], "summary": "Connected spin double cover of SO₀(2,4); not its universal cover. The complex 4 × 4 matrices A with det A = 1 and A†HA = H, where H = diag(1,1,−1,−1).", "definition": "The complex 4 × 4 matrices A with det A = 1 and A†HA = H, where H = diag(1,1,−1,−1).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₄", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: S(U(2) × U(2))", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Connected spin double cover of SO₀(2,4); not its universal cover", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(2,2) ≅ so(2,4)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "S(U(2) × U(2))", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "The maximal compact subgroup is S(U(2) × U(2)), whose determinant winding gives fundamental group Z. The scalar center is C₄. The orthogonal action on a real form of Λ²C⁴ gives a double cover of SO₀(2,4) with kernel {±I}; SU(2,2) itself is not simply connected.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The defining complex 4 is a chiral spinor for the connected orthogonal cover. Its exterior square has an invariant real structure and gives the real vector 6 of signature (2,4). Because −I acts nontrivially on the 4, that 4 does not descend to SO₀(2,4).", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11: exterior-square construction and double-cover kernel" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:su-2-2-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:su-2-2-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:su-2-2-to-so-2-4", "label": "Explicit homomorphism", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 15, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:su-2-2-r" } }, { "id": "group:sp-4-r", "slug": "sp-4-r", "name": "Real symplectic group on a four-dimensional space", "symbol": "Sp(4, R)", "category": "lie", "level": "reference", "aliases": [ "Spin₀(2,3)", "Spin₀(3,2)" ], "searchTerms": [ "sp(4, R) ≅ so(2,3)", "real rank 2" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Global form" ], "summary": "Connected spin double cover of SO₀(2,3); not its universal cover. The real 4 × 4 matrices A satisfying AᵀJA = J for a fixed nondegenerate alternating matrix J.", "definition": "The real 4 × 4 matrices A satisfying AᵀJA = J for a fixed nondegenerate alternating matrix J.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: U(2)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Connected spin double cover of SO₀(2,3); not its universal cover", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sp(4, R) ≅ so(2,3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "U(2)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Cartan decomposition retracts Sp(4,R) onto U(2), giving fundamental group Z. Its center is {±I}. Its orthogonal five-dimensional action is a double cover of SO₀(2,3), whose fundamental group is Z × C₂. This real group is different from compact Sp(2).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The real defining 4 preserves its symplectic form. The primitive part of Λ²R⁴ has dimension 5, carries an invariant quadratic form of signature (2,3) up to overall sign, and gives the double-cover map.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: five-dimensional orthogonal construction for the real rank-two symplectic group" } ], "assumptions": [] } ] }, { "id": "notation", "title": "Symplectic notation", "paragraphs": [ { "text": "This site writes compact Sp(n) for quaternionic n × n unitary matrices, also called USp(2n). It writes Sp(2n,R) for real matrices of size 2n preserving an alternating form. The two conventions count different dimensions.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sp-4-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sp-4-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:sp-4-r-to-so-2-3", "label": "Explicit homomorphism", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 10, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:sp-4-r" } }, { "id": "group:so-1-3-identity", "slug": "so-1-3-identity", "name": "Proper orthochronous Lorentz group in four dimensions", "symbol": "SO₀(1,3)", "category": "lie", "level": "reference", "aliases": [ "SO⁺(1,3)", "SO₀(3,1)", "Proper orthochronous Lorentz group" ], "searchTerms": [ "sl(2, C) viewed over R", "real rank 1" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Global form" ], "summary": "SL(2,C)/{±I}. The identity component of the real determinant-one matrices preserving a quadratic form with signature (1,3).", "definition": "The identity component of the real determinant-one matrices preserving a quadratic form with signature (1,3).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(3)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "SL(2,C)/{±I}", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) covering SO(3,1)" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(2, C) viewed over R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Its maximal compact subgroup is SO(3), and its manifold is SO(3) × R³. The simply connected double cover is SL(2,C), acting on Hermitian 2 × 2 matrices by X ↦ AXA†.", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) covering SO(3,1)" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "Finite-dimensional complex irreducibles of the SL(2,C) cover have labels (jL,jR). They descend to SO₀(1,3) exactly when jL + jR is an integer. The two Weyl spinors therefore belong to the cover, while the vector (1/2,1/2) descends.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "knapp-structure", "locator": "Theorem 6.105: complex simple algebras viewed as real" } ], "derivation": { "method": "The complexified Lorentz algebra is sl(2,C) ⊕ sl(2,C). The covering element −I acts with product sign (−1)^(2jL+2jR).", "inputs": [ "algebra:sl-2-c-as-real", "SL(2,C) → SO₀(1,3) has kernel {±I}" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sl-2-c-as-real", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sl-2-c-as-real", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:sl-2-c-to-lorentz", "label": "Explicit homomorphism", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) covering SO(3,1)" } ] }, { "type": "covered-by", "target": "group:sl-2-c", "label": "Specified connected covering group", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) covering SO(3,1)" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 6, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:sl-2-c-as-real" } }, { "id": "group:so-2-4-identity", "slug": "so-2-4-identity", "name": "Identity component of the orthogonal group of signature (2,4)", "symbol": "SO₀(2,4)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "su(2,2) ≅ so(2,4)", "real rank 2" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Global form" ], "summary": "SU(2,2)/{±I}. The identity component of the determinant-one real 6 × 6 matrices preserving a quadratic form with signature (2,4).", "definition": "The identity component of the determinant-one real 6 × 6 matrices preserving a quadratic form with signature (2,4).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "Z × C₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(2) × SO(4)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "SU(2,2)/{±I}", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.11: SU(2,2) double cover" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(2,2) ≅ so(2,4)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(2) × SO(4)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "A Cartan decomposition gives maximal compact subgroup SO(2) × SO(4), hence π₁ = Z × C₂. The matrix −I lies in the identity component because both signature blocks have even dimension; the center is {±I}. Its spin double cover SU(2,2) still has π₁ = Z.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.11: SU(2,2) double cover" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The vector 6 is an ordinary representation of this matrix group. The chiral spinor 4 of SU(2,2) does not descend through the order-two covering kernel. For a spacetime application, specify separately whether the action uses this group, its projective quotient, or a further covering group.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11: exterior-square orthogonal action and its kernel" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:su-2-2-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:su-2-2-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:su-2-2-to-so-2-4", "label": "Explicit homomorphism", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.11: SU(2,2) double cover" } ] }, { "type": "covered-by", "target": "group:su-2-2", "label": "Specified connected covering group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.11: SU(2,2) double cover" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 15, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:su-2-2-r" } }, { "id": "group:so-2-3-identity", "slug": "so-2-3-identity", "name": "Identity component of the orthogonal group of signature (2,3)", "symbol": "SO₀(2,3)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "sp(4, R) ≅ so(2,3)", "real rank 2" ], "tags": [ "Noncompact Lie group", "Simple algebra", "Global form" ], "summary": "Sp(4,R)/{±I}. The identity component of the determinant-one real 5 × 5 matrices preserving a quadratic form with signature (2,3).", "definition": "The identity component of the determinant-one real 5 × 5 matrices preserving a quadratic form with signature (2,3).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [] }, { "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "Z × C₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(2) × SO(3)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [] }, { "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [] }, { "key": "global-form", "label": "Global form", "value": "Sp(4,R)/{±I}", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic double cover of SO(3,2)" } ], "assumptions": [] }, { "key": "lie-algebra", "label": "Real Lie algebra", "value": "sp(4, R) ≅ so(2,3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [] }, { "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(2) × SO(3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [] } ], "sections": [ { "id": "topology", "title": "Global structure and topology", "paragraphs": [ { "text": "Its maximal compact subgroup is SO(2) × SO(3), giving π₁ = Z × C₂. The connected spin cover is Sp(4,R), whose fundamental group is Z. Neither the word Spin nor this double cover makes that covering group simply connected.", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic double cover of SO(3,2)" } ], "assumptions": [] } ] }, { "id": "representations", "title": "Representation category and descent", "paragraphs": [ { "text": "The vector 5 is an ordinary representation of SO₀(2,3). The symplectic 4 belongs to its connected spin cover Sp(4,R), since the covering kernel acts by minus the identity.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8: four-dimensional defining module and five-dimensional orthogonal action" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:sp-4-r", "label": "Real Lie algebra", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-real-form", "target": "real-form:sp-4-r", "label": "Real-form context", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ] }, { "type": "has-map", "target": "map:sp-4-r-to-so-2-3", "label": "Explicit homomorphism", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic double cover of SO(3,2)" } ] }, { "type": "covered-by", "target": "group:sp-4-r", "label": "Specified connected covering group", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic double cover of SO(3,2)" } ] } ], "capabilities": [], "kind": "group", "data": { "category": "lie", "dimension": 10, "compact": false, "connected": true, "abelian": false, "algebraId": "algebra:sp-4-r" } }, { "id": "map:su-2-to-so-3", "slug": "su-2-to-so-3", "name": "Spin cover of three-dimensional rotations", "symbol": "SU(2) → SO(3)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product. Kernel: {±I} ≅ C₂.", "definition": "A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.", "definitionEvidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "su-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-3", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO(3)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "This is the universal covering map. A spin-j SU(2) representation descends along it exactly when 2j is even. With compatible axis conventions, diag(eⁱθ,e⁻ⁱθ) maps to a rotation through 2θ.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The torus angle in SU(2) has period 2π, but the image rotation has angle 2θ. Tracking that factor is necessary when comparing the two spherical quotient fibrations.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "garrett-isogenies", "locator": "Section 2.1: adjoint rotation action" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:su-2", "label": "Domain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ] }, { "type": "has-codomain", "target": "group:so-3", "label": "Codomain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ] } ], "capabilities": [ "global-forms" ], "kind": "map", "data": { "domain": "group:su-2", "codomain": "group:so-3", "kernel": "{±I} ≅ C₂", "image": "SO(3)", "injective": false, "surjective": true, "capabilityPresets": { "global-forms": { "group": "so3" } } } }, { "id": "map:su-3-to-psu-3", "slug": "su-3-to-psu-3", "name": "Adjoint quotient of SU(3)", "symbol": "SU(3) → PSU(3)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "A ↦ its coset in SU(3)/C₃. Kernel: {exp(2πik/3)I : k = 0,1,2} ≅ C₃.", "definition": "A ↦ its coset in SU(3)/C₃.", "definitionEvidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "su-3", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "psu-3", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "A ↦ its coset in SU(3)/C₃.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{exp(2πik/3)I : k = 0,1,2} ≅ C₃", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "PSU(3)", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "3", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The defining 3 is not a representation of PSU(3). A highest weight (p,q) descends precisely when p + 2q ≡ 0 mod 3.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:su-3", "label": "Domain group", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ] }, { "type": "has-codomain", "target": "group:psu-3", "label": "Codomain group", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ] } ], "capabilities": [ "global-forms" ], "kind": "map", "data": { "domain": "group:su-3", "codomain": "group:psu-3", "kernel": "{exp(2πik/3)I : k = 0,1,2} ≅ C₃", "image": "PSU(3)", "injective": false, "surjective": true, "capabilityPresets": { "global-forms": { "group": "psu3" } } } }, { "id": "map:su-2-to-su-3-regular", "slug": "su-2-to-su-3-regular", "name": "Regular block inclusion in SU(3)", "symbol": "SU(2) ↪ SU(3)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [], "tags": [ "Embedding" ], "summary": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed. Kernel: {I}.", "definition": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.", "definitionEvidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "su-2", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "su-3", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{I}", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "{diag(A,1) : A ∈ SU(2)} ≅ SU(2)", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "Yes", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "No", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The SU(3) fundamental branches as 3 → 2 ⊕ 1. The adjoint branches as 8 → 3 ⊕ 2 ⊕ 2 ⊕ 1.", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "For the adjoint rule, restrict End(C³) and remove its scalar summand. End(2 ⊕ 1) decomposes as (3 ⊕ 1) ⊕ 2 ⊕ 2 ⊕ 1, giving the stated eight-dimensional result after subtracting one singlet.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "haber", "locator": "Problem 5: regular inclusion" } ], "derivation": { "method": "Restrict End(3) − 1 using the specified block inclusion and the SU(2) tensor rule 2⊗2=3⊕1.", "inputs": [ "3 restricts to 2⊕1", "Adjoint SU(3) = End(3) minus scalars" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:su-2", "label": "Domain group", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ] }, { "type": "has-codomain", "target": "group:su-3", "label": "Codomain group", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ] } ], "capabilities": [ "branching", "su3-branching" ], "kind": "map", "data": { "domain": "group:su-2", "codomain": "group:su-3", "kernel": "{I}", "image": "{diag(A,1) : A ∈ SU(2)} ≅ SU(2)", "injective": true, "surjective": false, "capabilityPresets": { "branching": { "embedding": "regular", "rep": "8" }, "su3-branching": { "labels": "1,1", "map": "regular" } } } }, { "id": "map:su-2-to-su-3-principal", "slug": "su-2-to-su-3-principal", "name": "Principal spin-one homomorphism into SU(3)", "symbol": "SU(2) → SU(3), spin 1", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [], "tags": [ "Noninjective homomorphism" ], "summary": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3). Kernel: {±I} ≅ C₂.", "definition": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).", "definitionEvidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "su-2", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "su-3", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "An SU(3)-conjugate of the real SO(3) subgroup", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "No", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The fundamental branches as 3 → 3, and the adjoint as 8 → 3 ⊕ 5. This is a Lie-algebra embedding, but the group homomorphism is not injective.", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The adjoint rule follows from 3 ⊗ 3 = 1 ⊕ 3 ⊕ 5 after removing the scalar endomorphism. Every surviving summand has integer spin, consistent with the central kernel.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" } ], "derivation": { "method": "Use the SU(2) spin-one tensor decomposition and Adj(SU(3)) = End(3) − 1.", "inputs": [ "The defining SU(3) representation restricts to spin 1", "Spin 1 ⊗ spin 1 = spin 0 ⊕ spin 1 ⊕ spin 2" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:su-2", "label": "Domain group", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ] }, { "type": "has-codomain", "target": "group:su-3", "label": "Codomain group", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ] } ], "capabilities": [ "branching", "su3-branching" ], "kind": "map", "data": { "domain": "group:su-2", "codomain": "group:su-3", "kernel": "{±I} ≅ C₂", "image": "An SU(3)-conjugate of the real SO(3) subgroup", "injective": false, "surjective": false, "capabilityPresets": { "branching": { "embedding": "principal", "rep": "8" }, "su3-branching": { "labels": "1,1", "map": "principal" } } } }, { "id": "map:u-1-to-su-2", "slug": "u-1-to-su-2", "name": "Normalized maximal torus of SU(2)", "symbol": "U(1) ↪ SU(2)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [], "tags": [ "Embedding" ], "summary": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π. Kernel: {1}.", "definition": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "su-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{1}", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "{diag(z,z⁻¹) : |z|=1}", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The defining doublet has U(1) weights +1 and −1 in this convention. Its homogeneous quotient is SU(2)/U(1) ≅ S².", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "The induced map π₁(U(1)) → π₁(SU(2)) is zero. In the quotient fibration, the boundary map π₂(S²) → π₁(U(1)) is multiplication by +1 when the sphere generator is oriented to match this fiber generator.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Apply the exact sequence to U(1) → SU(2) → S², using π₂(S³)=π₁(S³)=0.", "inputs": [ "SU(2) is S³", "π₁(U(1))=Z", "Fiber loop θ:0→2π", "Choice of positively compatible sphere generator" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:u-1", "label": "Domain group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "has-codomain", "target": "group:su-2", "label": "Codomain group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "defines-quotient", "target": "space:su-2-over-u-1", "label": "Associated homogeneous quotient", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] } ], "capabilities": [ "topology" ], "kind": "map", "data": { "domain": "group:u-1", "codomain": "group:su-2", "kernel": "{1}", "image": "{diag(z,z⁻¹) : |z|=1}", "injective": true, "surjective": false, "capabilityPresets": { "topology": { "kind": "su2" } } } }, { "id": "map:u-1-to-so-3", "slug": "u-1-to-so-3", "name": "Normalized axial rotation subgroup of SO(3)", "symbol": "U(1) ≅ SO(2) ↪ SO(3)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [], "tags": [ "Embedding" ], "summary": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ modulo 2π. Kernel: {1}.", "definition": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ modulo 2π.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ modulo 2π.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{1}", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "Rotations fixing a chosen oriented axis, isomorphic to SO(2)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "This uses the actual spatial rotation angle θ. The quotient SO(3)/SO(2) is S². The inclusion sends a winding number to its parity in π₁(SO(3)) = C₂.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "A 2π rotation lifts to a path from I to −I in SU(2), whereas a 4π rotation lifts to a loop. Therefore the inclusion induces reduction modulo two, and the boundary map π₂(S²) → π₁(SO(2)) is multiplication by +2 after choosing compatible orientations.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "The kernel of Z → C₂ is 2Z. Since π₂(SO(3))=0, the boundary map is injective with this image.", "inputs": [ "map:su-2-to-so-3", "SO(3) is RP³", "π₂(RP³)=π₂(S³)=0", "Fiber loop is one spatial 2π rotation" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:u-1", "label": "Domain group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] }, { "type": "has-codomain", "target": "group:so-3", "label": "Codomain group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] }, { "type": "defines-quotient", "target": "space:so-3-over-so-2", "label": "Associated homogeneous quotient", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] } ], "capabilities": [ "topology" ], "kind": "map", "data": { "domain": "group:u-1", "codomain": "group:so-3", "kernel": "{1}", "image": "Rotations fixing a chosen oriented axis, isomorphic to SO(2)", "injective": true, "surjective": false, "capabilityPresets": { "topology": { "kind": "so3" } } } }, { "id": "map:spin-4-to-so-4", "slug": "spin-4-to-so-4", "name": "Quaternionic spin cover in four Euclidean dimensions", "symbol": "Spin(4) → SO(4)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b. Kernel: {(1,1),(−1,−1)} ≅ C₂.", "definition": "(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b.", "definitionEvidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "spin-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{(1,1),(−1,−1)} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO(4)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The kernel is the diagonal order-two subgroup, not the full C₂ × C₂ center of Spin(4). Hence (jL,jR) descends exactly when jL + jR is an integer.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:spin-4", "label": "Domain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ] }, { "type": "has-codomain", "target": "group:so-4", "label": "Codomain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:spin-4", "codomain": "group:so-4", "kernel": "{(1,1),(−1,−1)} ≅ C₂", "image": "SO(4)", "injective": false, "surjective": true } }, { "id": "map:su-4-to-so-6", "slug": "su-4-to-so-6", "name": "Exterior-square realization of the Spin(6) cover", "symbol": "SU(4) → SO(6)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form. Kernel: {±I} ≅ C₂.", "definition": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.", "definitionEvidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "su-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-6", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO(6)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "This identifies SU(4) with Spin(6). The central element iI acts by −I on the vector 6, while −I acts trivially; quotienting by the full C₄ center would give PSO(6).", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:su-4", "label": "Domain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ] }, { "type": "has-codomain", "target": "group:so-6", "label": "Codomain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:su-4", "codomain": "group:so-6", "kernel": "{±I} ≅ C₂", "image": "SO(6)", "injective": false, "surjective": true } }, { "id": "map:su-4-to-psu-4", "slug": "su-4-to-psu-4", "name": "Full central quotient of SU(4)", "symbol": "SU(4) → PSU(4)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "A ↦ its coset in SU(4)/C₄. Kernel: {I,iI,−I,−iI} ≅ C₄.", "definition": "A ↦ its coset in SU(4)/C₄.", "definitionEvidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "su-4", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "psu-4", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "A ↦ its coset in SU(4)/C₄.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{I,iI,−I,−iI} ≅ C₄", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "PSU(4)", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "4", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "This is the full adjoint quotient. It has a different kernel from the vector-cover map SU(4) → SO(6).", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:su-4", "label": "Domain group", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] }, { "type": "has-codomain", "target": "group:psu-4", "label": "Codomain group", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:su-4", "codomain": "group:psu-4", "kernel": "{I,iI,−I,−iI} ≅ C₄", "image": "PSU(4)", "injective": false, "surjective": true } }, { "id": "map:sp-2-to-so-5", "slug": "sp-2-to-so-5", "name": "Quaternionic construction of the Spin(5) cover", "symbol": "Sp(2) → SO(5)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign. Kernel: {±I} ≅ C₂.", "definition": "A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign.", "definitionEvidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "sp-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-5", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO(5)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The quaternionic compact convention is essential: Sp(2) here is USp(4), not Sp(4,R). The map exhibits Sp(2) ≅ Spin(5).", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:sp-2", "label": "Domain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ] }, { "type": "has-codomain", "target": "group:so-5", "label": "Codomain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:sp-2", "codomain": "group:so-5", "kernel": "{±I} ≅ C₂", "image": "SO(5)", "injective": false, "surjective": true } }, { "id": "map:spin-8-to-so-8", "slug": "spin-8-to-so-8", "name": "Vector spin cover in eight Euclidean dimensions", "symbol": "Spin(8) → SO(8)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8). Kernel: {±1} ≅ C₂.", "definition": "g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8).", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "spin-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8).", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±1} ≅ C₂", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO(8)", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "This specifies the vector-cover kernel inside the full C₂ × C₂ center. Triality changes the role of vector and half-spin modules and permutes the three order-two kernels.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "Although the three quotients by order-two central subgroups are abstractly isomorphic, a representation descends according to the kernel of the specified map. The quotient name alone does not retain this choice.", "evidence": [ { "source": "conrad", "locator": "Remark T.2.1, printed page 262: D₄ triality and quotient groups" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:spin-8", "label": "Domain group", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] }, { "type": "has-codomain", "target": "group:so-8", "label": "Codomain group", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:spin-8", "codomain": "group:so-8", "kernel": "{±1} ≅ C₂", "image": "SO(8)", "injective": false, "surjective": true } }, { "id": "map:spin-10-to-so-10", "slug": "spin-10-to-so-10", "name": "Vector spin cover in ten Euclidean dimensions", "symbol": "Spin(10) → SO(10)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "g ↦ (v ↦ gvg⁻¹) on R¹⁰ inside its Euclidean Clifford algebra. Kernel: {±1} ≅ C₂.", "definition": "g ↦ (v ↦ gvg⁻¹) on R¹⁰ inside its Euclidean Clifford algebra.", "definitionEvidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "spin-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "g ↦ (v ↦ gvg⁻¹) on R¹⁰ inside its Euclidean Clifford algebra.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±1} ≅ C₂", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO(10)", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The full center of Spin(10) is C₄, while this map has the unique order-two kernel. The half-spin 16s do not descend through it.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:spin-10", "label": "Domain group", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] }, { "type": "has-codomain", "target": "group:so-10", "label": "Codomain group", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:spin-10", "codomain": "group:so-10", "kernel": "{±1} ≅ C₂", "image": "SO(10)", "injective": false, "surjective": true } }, { "id": "map:sl-2-c-to-lorentz", "slug": "sl-2-c-to-lorentz", "name": "Hermitian-matrix Lorentz spin cover", "symbol": "SL(2, C) → SO₀(1,3)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved. Kernel: {±I} ≅ C₂.", "definition": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.", "definitionEvidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "sl-2-c", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-1-3-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO₀(1,3)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The connected source maps onto the identity component of the Lorentz group. It does not cover time reversal or spatial parity components. SL(2,C) is simply connected as a real Lie group.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "Preservation of det X follows from det A = 1. If AXA† = X for every Hermitian X, then A is scalar and equals ±I. The derivative is an isomorphism of six-dimensional real Lie algebras; connectedness identifies the image as SO₀(1,3).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "derivation": { "method": "Check the determinant, central kernel, derivative, and identity component directly from the displayed action.", "inputs": [ "det A=1", "Hermitian 2×2 matrices form a real four-dimensional vector space", "SL(2,C) is connected" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:sl-2-c", "label": "Domain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ] }, { "type": "has-codomain", "target": "group:so-1-3-identity", "label": "Codomain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:sl-2-c", "codomain": "group:so-1-3-identity", "kernel": "{±I} ≅ C₂", "image": "SO₀(1,3)", "injective": false, "surjective": true } }, { "id": "map:sl-2-r-to-psl-2-r", "slug": "sl-2-r-to-psl-2-r", "name": "Adjoint central quotient of SL(2,R)", "symbol": "SL(2, R) → PSL(2, R)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form. Kernel: {±I} ≅ C₂.", "definition": "A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form.", "definitionEvidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "sl-2-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "psl-2-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "PSL(2,R) ≅ SO₀(2,1)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "Both groups have infinite cyclic fundamental group. With compatible circle generators, this double cover induces multiplication by two on π₁; neither group is the universal cover.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:sl-2-r", "label": "Domain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ] }, { "type": "has-codomain", "target": "group:psl-2-r", "label": "Codomain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:sl-2-r", "codomain": "group:psl-2-r", "kernel": "{±I} ≅ C₂", "image": "PSL(2,R) ≅ SO₀(2,1)", "injective": false, "surjective": true } }, { "id": "map:universal-sl-2-r-to-sl-2-r", "slug": "universal-sl-2-r-to-sl-2-r", "name": "Universal covering map of SL(2,R)", "symbol": "Universal SL(2,R) cover → SL(2,R)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism. Kernel: ⟨z²⟩ ≅ Z, where z lifts a π rotation.", "definition": "Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "universal-cover-sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "⟨z²⟩ ≅ Z, where z lifts a π rotation", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SL(2,R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "Infinitely many sheets", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The center of the domain is ⟨z⟩. Its subgroup ⟨z²⟩ is the kernel of this projection; composing with SL(2,R) → PSL(2,R) enlarges the kernel to the full center.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "Choose R(θ) as the standard planar rotation matrix inside SL(2,R). The path θ:0→π ends at −I and defines z; the path θ:0→2π is one fundamental loop in SL(2,R) and defines z².", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Lift the maximal compact rotation subgroup through the universal covering map.", "inputs": [ "SL(2,R) retracts to SO(2)", "π₁(SO(2)) is generated by θ:0→2π", "R(π)=−I" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:universal-cover-sl-2-r", "label": "Domain group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ] }, { "type": "has-codomain", "target": "group:sl-2-r", "label": "Codomain group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:universal-cover-sl-2-r", "codomain": "group:sl-2-r", "kernel": "⟨z²⟩ ≅ Z, where z lifts a π rotation", "image": "SL(2,R)", "injective": false, "surjective": true } }, { "id": "map:su-2-2-to-so-2-4", "slug": "su-2-2-to-so-2-4", "name": "Exterior-square spin cover in signature (2,4)", "symbol": "SU(2,2) → SO₀(2,4)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign. Kernel: {±I} ≅ C₂.", "definition": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.", "definitionEvidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "su-2-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-2-4-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO₀(2,4)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "This map identifies SU(2,2) with the connected spin double cover. Its source has π₁ = Z, so this is not a universal covering map. A further quotient of the target by {±I} is a different global group.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:su-2-2", "label": "Domain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ] }, { "type": "has-codomain", "target": "group:so-2-4-identity", "label": "Codomain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:su-2-2", "codomain": "group:so-2-4-identity", "kernel": "{±I} ≅ C₂", "image": "SO₀(2,4)", "injective": false, "surjective": true } }, { "id": "map:sp-4-r-to-so-2-3", "slug": "sp-4-r-to-so-2-3", "name": "Primitive-exterior-square spin cover in signature (2,3)", "symbol": "Sp(4, R) → SO₀(2,3)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [], "tags": [ "Covering or central quotient" ], "summary": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign. Kernel: {±I} ≅ C₂.", "definition": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.", "definitionEvidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "properties": [ { "key": "domain", "label": "Domain", "value": "sp-4-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [] }, { "key": "codomain", "label": "Codomain", "value": "so-2-3-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [] }, { "key": "formula", "label": "Specified homomorphism", "value": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [] }, { "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "SO₀(2,3)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [] }, { "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [] } ], "sections": [ { "id": "meaning", "title": "Global and representation consequences", "paragraphs": [ { "text": "The source has maximal compact U(2) and fundamental group Z; the target has maximal compact SO(2) × SO(3) and fundamental group Z × C₂. The covering is double but is not universal.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [] }, { "text": "On maximal compact subgroups, the covering may be written U(2) → SO(2) × SO(3) using the determinant and the projective SU(2) action. A generator of π₁(U(2)) maps to (1,1 mod 2), an index-two subgroup of Z × C₂.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "derivation": { "method": "Represent a generator by diag(eⁱᵗ,1), 0≤t≤2π. Its determinant winds once and its SU(2) lift ends at −I, giving the nontrivial SO(3) loop.", "inputs": [ "U(2)=(SU(2)×U(1))/diagonal C₂", "SU(2)→SO(3) has kernel {±I}", "The determinant map induces π₁(U(2))≅Z" ] }, "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:sp-4-r", "label": "Domain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ] }, { "type": "has-codomain", "target": "group:so-2-3-identity", "label": "Codomain group", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:sp-4-r", "codomain": "group:so-2-3-identity", "kernel": "{±I} ≅ C₂", "image": "SO₀(2,3)", "injective": false, "surjective": true } }, { "id": "space:su-2-over-u-1", "slug": "su-2-over-u-1", "name": "Hopf sphere as SU(2)/U(1)", "symbol": "SU(2)/U(1) ≅ S²", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [], "tags": [ "Homogeneous space", "Sphere", "Exact homotopy sequence" ], "summary": "The sphere quotient with its specified U(1) embedding and boundary map of winding one.", "definition": "The right-coset homogeneous space of SU(2) by the embedded torus eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ).", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "manifold", "label": "Manifold", "value": "S²", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [] }, { "key": "pi2", "label": "Second homotopy group π₂", "value": "Z", "evidence": [ { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [] }, { "key": "fiber-period", "label": "Fiber angle period", "value": "2π", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "boundary-map", "label": "Boundary π₂(S²) → π₁(U(1))", "value": "n ↦ n", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Exactness of 0 → π₂(S²) → π₁(U(1)) → 0 forces the boundary to be an isomorphism; orientation fixes its positive sign.", "inputs": [ "SU(2)=S³", "π₂(S³)=0", "π₁(S³)=0", "π₁(U(1))=Z" ] }, "assumptions": [ "The U(1) generator is θ:0→2π in diag(eⁱθ,e⁻ⁱθ).", "Choose the sphere generator so its boundary is the positive fiber generator." ] } ], "sections": [ { "id": "sequence", "title": "The exact sequence with generators", "paragraphs": [ { "text": "The relevant segment is 0 → Z → Z → 0. The two displayed Z groups are respectively π₂(S²) and π₁(U(1)). With the generator convention stated here, the first nonzero arrow is multiplication by +1.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Apply the fibration homotopy sequence to U(1) → S³ → S².", "inputs": [ "map:u-1-to-su-2", "π₂(S³)=π₁(S³)=0" ] }, "assumptions": [] }, { "text": "The same base manifold occurs for SO(3)/SO(2), but the embedding and the total group change the boundary homomorphism. The integer here is a fiber winding; converting it to a physical magnetic charge also requires the generator and coupling convention.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "quotient-of", "target": "group:su-2", "label": "Total group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "has-isotropy", "target": "group:u-1", "label": "Embedded torus", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "defined-by", "target": "map:u-1-to-su-2", "label": "Normalized subgroup map", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "same-manifold-as", "target": "space:so-3-over-so-2", "label": "Same S², different fibration", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] } ], "capabilities": [ "topology" ], "kind": "space", "data": { "construction": "Right-coset quotient by the explicitly embedded maximal torus", "groupId": "group:su-2", "subgroupId": "group:u-1", "mapId": "map:u-1-to-su-2", "capabilityPresets": { "topology": { "kind": "su2" } } } }, { "id": "space:so-3-over-so-2", "slug": "so-3-over-so-2", "name": "Rotation sphere as SO(3)/SO(2)", "symbol": "SO(3)/SO(2) ≅ S²", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [], "tags": [ "Homogeneous space", "Sphere", "Exact homotopy sequence" ], "summary": "The sphere quotient with its axial SO(2) embedding and boundary map of winding two.", "definition": "The homogeneous space of rotations modulo the subgroup fixing a chosen axis, with SO(2) identified with U(1) by the actual spatial rotation angle.", "definitionEvidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "manifold", "label": "Manifold", "value": "S²", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [] }, { "key": "pi2", "label": "Second homotopy group π₂", "value": "Z", "evidence": [ { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [] }, { "key": "fiber-period", "label": "Physical rotation angle period", "value": "2π", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] }, { "key": "inclusion-on-pi1", "label": "Inclusion π₁(SO(2)) → π₁(SO(3))", "value": "n ↦ n mod 2", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "derivation": { "method": "A 2π spatial rotation lifts from I to −I in SU(2), while a 4π rotation closes.", "inputs": [ "map:su-2-to-so-3", "Fiber angle is the spatial angle θ with period 2π" ] }, "assumptions": [] }, { "key": "boundary-map", "label": "Boundary π₂(S²) → π₁(SO(2))", "value": "n ↦ 2n", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Exactness identifies the boundary image with the kernel 2Z of Z→C₂. The boundary is injective because π₂(SO(3))=0.", "inputs": [ "SO(3)=RP³", "π₂(RP³)=π₂(S³)=0", "π₁(SO(3))=C₂", "Inclusion sends winding to its parity" ] }, "assumptions": [ "The fiber generator is one positive spatial 2π rotation.", "Choose the sphere generator to make the boundary coefficient positive." ] } ], "sections": [ { "id": "sequence", "title": "The exact sequence with generators", "paragraphs": [ { "text": "The relevant segment is 0 → Z → Z → C₂ → 0. The arrow between the two infinite cyclic groups is multiplication by +2, and the following arrow reduces an integer modulo two.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Use the fibration SO(2) → SO(3) → S² and the induced map on fundamental groups.", "inputs": [ "map:u-1-to-so-3", "π₂(SO(3))=0", "π₁(SO(3))=C₂", "π₁(S²)=0" ] }, "assumptions": [] }, { "text": "The subgroup record is U(1), used through the specified isomorphism U(1) ≅ SO(2). This avoids creating a second abstract circle-group record while retaining its concrete rotational action.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "quotient-of", "target": "group:so-3", "label": "Total group", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "has-isotropy", "target": "group:u-1", "label": "Circle realized as SO(2)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "defined-by", "target": "map:u-1-to-so-3", "label": "Normalized rotational subgroup map", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] }, { "type": "same-manifold-as", "target": "space:su-2-over-u-1", "label": "Same S², different fibration", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ] } ], "capabilities": [ "topology" ], "kind": "space", "data": { "construction": "Right-coset quotient by the axial rotation subgroup SO(2)", "groupId": "group:so-3", "subgroupId": "group:u-1", "mapId": "map:u-1-to-so-3", "capabilityPresets": { "topology": { "kind": "so3" } } } }, { "id": "representation:c-3-1", "slug": "c-3-1", "name": "C₃ irreducible representation 1", "symbol": "C₃ · 1", "category": "finite", "level": "executable", "aliases": [ "C₃ irrep 1" ], "searchTerms": [ "c3", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of C₃, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1 of C₃, specified by its character on the named conjugacy classes. Trivial representation", "definitionEvidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. The two nonidentity classes are the chosen generator r and its square.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:c-3", "label": "Representation domain", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:c-3", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1", "displayTex": "C_3:\\quad \\mathbf{1}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "3a", "label": "r", "latex": "r", "labelTeX": "r", "size": "1", "order": 3, "representative": "r" }, { "id": "3b", "label": "r²", "latex": "r^2", "labelTeX": "r^2", "size": "1", "order": 3, "representative": "r^2" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" } ] }, "capabilityPresets": { "finite-characters": { "group": "c3", "operation": "tensor", "left": "1", "right": "1", "copies": "1", "degree": "2" } } } }, { "id": "representation:c-3-chi", "slug": "c-3-chi", "name": "C₃ irreducible representation χ", "symbol": "C₃ · χ", "category": "finite", "level": "executable", "aliases": [ "C₃ irrep χ" ], "searchTerms": [ "c3", "Complex", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of C₃, of complex type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled χ of C₃, specified by its character on the named conjugacy classes. The chosen generator acts by omega", "definitionEvidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Complex", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "0", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "χ²", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. The two nonidentity classes are the chosen generator r and its square.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; complex describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:c-3", "label": "Representation domain", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] }, { "type": "conjugate", "target": "representation:c-3-chi2", "label": "Complex conjugate representation", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:c-3", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "chi", "displayTex": "C_3:\\quad \\chi", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "3a", "label": "r", "latex": "r", "labelTeX": "r", "size": "1", "order": 3, "representative": "r" }, { "id": "3b", "label": "r²", "latex": "r^2", "labelTeX": "r^2", "size": "1", "order": 3, "representative": "r^2" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "ω", "latex": "\\omega", "coefficients": [ "0", "1" ], "basis": "1,omega" }, { "text": "ω²", "latex": "\\omega^2", "coefficients": [ "-1", "-1" ], "basis": "1,omega" } ] }, "capabilityPresets": { "finite-characters": { "group": "c3", "operation": "tensor", "left": "chi", "right": "chi2", "copies": "1", "degree": "2" } } } }, { "id": "representation:c-3-chi2", "slug": "c-3-chi2", "name": "C₃ irreducible representation χ²", "symbol": "C₃ · χ²", "category": "finite", "level": "executable", "aliases": [ "C₃ irrep χ²" ], "searchTerms": [ "c3", "Complex", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of C₃, of complex type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled χ² of C₃, specified by its character on the named conjugacy classes. The chosen generator acts by omega squared", "definitionEvidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Complex", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "0", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "χ", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. The two nonidentity classes are the chosen generator r and its square.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; complex describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:c-3", "label": "Representation domain", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] }, { "type": "conjugate", "target": "representation:c-3-chi", "label": "Complex conjugate representation", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:c-3", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "chi2", "displayTex": "C_3:\\quad \\chi^2", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "3a", "label": "r", "latex": "r", "labelTeX": "r", "size": "1", "order": 3, "representative": "r" }, { "id": "3b", "label": "r²", "latex": "r^2", "labelTeX": "r^2", "size": "1", "order": 3, "representative": "r^2" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "ω²", "latex": "\\omega^2", "coefficients": [ "-1", "-1" ], "basis": "1,omega" }, { "text": "ω", "latex": "\\omega", "coefficients": [ "0", "1" ], "basis": "1,omega" } ] }, "capabilityPresets": { "finite-characters": { "group": "c3", "operation": "tensor", "left": "chi2", "right": "chi", "copies": "1", "degree": "2" } } } }, { "id": "representation:s-3-1", "slug": "s-3-1", "name": "S₃ irreducible representation 1", "symbol": "S₃ · 1", "category": "finite", "level": "executable", "aliases": [ "S₃ irrep 1" ], "searchTerms": [ "s3", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of S₃, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1 of S₃, specified by its character on the named conjugacy classes. Trivial representation", "definitionEvidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, transpositions, and three-cycles.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:s-3", "label": "Representation domain", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:s-3", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1", "displayTex": "S_3:\\quad \\mathbf{1}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)", "latex": "(12)", "labelTeX": "(12)", "size": "3", "order": 2, "representative": "(12)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "2", "order": 3, "representative": "(123)" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "s3", "operation": "tensor", "left": "1", "right": "1", "copies": "1", "degree": "2" } } } }, { "id": "representation:s-3-sgn", "slug": "s-3-sgn", "name": "S₃ irreducible representation sgn", "symbol": "S₃ · sgn", "category": "finite", "level": "executable", "aliases": [ "S₃ irrep sgn" ], "searchTerms": [ "s3", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of S₃, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled sgn of S₃, specified by its character on the named conjugacy classes. Sign of a permutation", "definitionEvidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "sgn", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, transpositions, and three-cycles.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:s-3", "label": "Representation domain", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:s-3", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "sgn", "displayTex": "S_3:\\quad \\mathrm{sgn}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)", "latex": "(12)", "labelTeX": "(12)", "size": "3", "order": 2, "representative": "(12)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "2", "order": 3, "representative": "(123)" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "s3", "operation": "tensor", "left": "sgn", "right": "sgn", "copies": "1", "degree": "2" } } } }, { "id": "representation:s-3-2", "slug": "s-3-2", "name": "S₃ irreducible representation 2", "symbol": "S₃ · 2", "category": "finite", "level": "executable", "aliases": [ "S₃ irrep 2" ], "searchTerms": [ "s3", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 2-dimensional ordinary complex irrep of S₃, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 2 of S₃, specified by its character on the named conjugacy classes. Standard representation: permutation representation minus the invariant line", "definitionEvidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "2", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, transpositions, and three-cycles.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:s-3", "label": "Representation domain", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:s-3", "category": "ordinary-complex", "field": "C", "dimension": 2, "labels": "2", "displayTex": "S_3:\\quad \\mathbf{2}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)", "latex": "(12)", "labelTeX": "(12)", "size": "3", "order": 2, "representative": "(12)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "2", "order": 3, "representative": "(123)" } ], "values": [ { "text": "2", "latex": "2", "coefficients": [ "2", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "s3", "operation": "tensor", "left": "2", "right": "2", "copies": "1", "degree": "2" } } } }, { "id": "representation:d-8-1", "slug": "d-8-1", "name": "D₈ irreducible representation 1", "symbol": "D₈ · 1", "category": "finite", "level": "executable", "aliases": [ "D₈ irrep 1" ], "searchTerms": [ "d8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of D₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1 of D₈, specified by its character on the named conjugacy classes. Trivial representation", "definitionEvidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:d-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:d-8", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1", "displayTex": "D_8:\\quad \\mathbf{1}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "r²", "latex": "r^2", "labelTeX": "r^2", "size": "1", "order": 2, "representative": "r^2" }, { "id": "4a", "label": "r", "latex": "r", "labelTeX": "r", "size": "2", "order": 4, "representative": "r" }, { "id": "2s", "label": "s", "latex": "s", "labelTeX": "s", "size": "2", "order": 2, "representative": "s" }, { "id": "2rs", "label": "rs", "latex": "rs", "labelTeX": "rs", "size": "2", "order": 2, "representative": "rs" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "d8", "operation": "tensor", "left": "1", "right": "1", "copies": "1", "degree": "2" } } } }, { "id": "representation:d-8-a", "slug": "d-8-a", "name": "D₈ irreducible representation a", "symbol": "D₈ · a", "category": "finite", "level": "executable", "aliases": [ "D₈ irrep a" ], "searchTerms": [ "d8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of D₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled a of D₈, specified by its character on the named conjugacy classes. r maps to +1 and s to -1; determinant of the doublet", "definitionEvidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "a", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:d-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:d-8", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "a", "displayTex": "D_8:\\quad a", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "r²", "latex": "r^2", "labelTeX": "r^2", "size": "1", "order": 2, "representative": "r^2" }, { "id": "4a", "label": "r", "latex": "r", "labelTeX": "r", "size": "2", "order": 4, "representative": "r" }, { "id": "2s", "label": "s", "latex": "s", "labelTeX": "s", "size": "2", "order": 2, "representative": "s" }, { "id": "2rs", "label": "rs", "latex": "rs", "labelTeX": "rs", "size": "2", "order": 2, "representative": "rs" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "d8", "operation": "tensor", "left": "a", "right": "a", "copies": "1", "degree": "2" } } } }, { "id": "representation:d-8-b", "slug": "d-8-b", "name": "D₈ irreducible representation b", "symbol": "D₈ · b", "category": "finite", "level": "executable", "aliases": [ "D₈ irrep b" ], "searchTerms": [ "d8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of D₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled b of D₈, specified by its character on the named conjugacy classes. r maps to -1 and s to +1", "definitionEvidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "b", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:d-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:d-8", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "b", "displayTex": "D_8:\\quad b", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "r²", "latex": "r^2", "labelTeX": "r^2", "size": "1", "order": 2, "representative": "r^2" }, { "id": "4a", "label": "r", "latex": "r", "labelTeX": "r", "size": "2", "order": 4, "representative": "r" }, { "id": "2s", "label": "s", "latex": "s", "labelTeX": "s", "size": "2", "order": 2, "representative": "s" }, { "id": "2rs", "label": "rs", "latex": "rs", "labelTeX": "rs", "size": "2", "order": 2, "representative": "rs" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "d8", "operation": "tensor", "left": "b", "right": "b", "copies": "1", "degree": "2" } } } }, { "id": "representation:d-8-c", "slug": "d-8-c", "name": "D₈ irreducible representation c", "symbol": "D₈ · c", "category": "finite", "level": "executable", "aliases": [ "D₈ irrep c" ], "searchTerms": [ "d8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of D₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled c of D₈, specified by its character on the named conjugacy classes. r maps to -1 and s to -1", "definitionEvidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "c", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:d-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:d-8", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "c", "displayTex": "D_8:\\quad c", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "r²", "latex": "r^2", "labelTeX": "r^2", "size": "1", "order": 2, "representative": "r^2" }, { "id": "4a", "label": "r", "latex": "r", "labelTeX": "r", "size": "2", "order": 4, "representative": "r" }, { "id": "2s", "label": "s", "latex": "s", "labelTeX": "s", "size": "2", "order": 2, "representative": "s" }, { "id": "2rs", "label": "rs", "latex": "rs", "labelTeX": "rs", "size": "2", "order": 2, "representative": "rs" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "d8", "operation": "tensor", "left": "c", "right": "c", "copies": "1", "degree": "2" } } } }, { "id": "representation:d-8-2", "slug": "d-8-2", "name": "D₈ irreducible representation 2", "symbol": "D₈ · 2", "category": "finite", "level": "executable", "aliases": [ "D₈ irrep 2" ], "searchTerms": [ "d8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 2-dimensional ordinary complex irrep of D₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 2 of D₈, specified by its character on the named conjugacy classes. Faithful real-type geometric doublet", "definitionEvidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "2", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:d-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:d-8", "category": "ordinary-complex", "field": "C", "dimension": 2, "labels": "2", "displayTex": "D_8:\\quad \\mathbf{2}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "r²", "latex": "r^2", "labelTeX": "r^2", "size": "1", "order": 2, "representative": "r^2" }, { "id": "4a", "label": "r", "latex": "r", "labelTeX": "r", "size": "2", "order": 4, "representative": "r" }, { "id": "2s", "label": "s", "latex": "s", "labelTeX": "s", "size": "2", "order": 2, "representative": "s" }, { "id": "2rs", "label": "rs", "latex": "rs", "labelTeX": "rs", "size": "2", "order": 2, "representative": "rs" } ], "values": [ { "text": "2", "latex": "2", "coefficients": [ "2", "0" ], "basis": "1" }, { "text": "-2", "latex": "-2", "coefficients": [ "-2", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "d8", "operation": "tensor", "left": "2", "right": "2", "copies": "1", "degree": "2" } } } }, { "id": "representation:q-8-1", "slug": "q-8-1", "name": "Q₈ irreducible representation 1", "symbol": "Q₈ · 1", "category": "finite", "level": "executable", "aliases": [ "Q₈ irrep 1" ], "searchTerms": [ "q8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of Q₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1 of Q₈, specified by its character on the named conjugacy classes. Trivial representation", "definitionEvidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:q-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:q-8", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1", "displayTex": "Q_8:\\quad \\mathbf{1}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "−1", "latex": "-1", "labelTeX": "-1", "size": "1", "order": 2, "representative": "-1" }, { "id": "4i", "label": "i", "latex": "i", "labelTeX": "i", "size": "2", "order": 4, "representative": "i" }, { "id": "4j", "label": "j", "latex": "j", "labelTeX": "j", "size": "2", "order": 4, "representative": "j" }, { "id": "4k", "label": "k", "latex": "k", "labelTeX": "k", "size": "2", "order": 4, "representative": "k" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "q8", "operation": "tensor", "left": "1", "right": "1", "copies": "1", "degree": "2" } } } }, { "id": "representation:q-8-a", "slug": "q-8-a", "name": "Q₈ irreducible representation a", "symbol": "Q₈ · a", "category": "finite", "level": "executable", "aliases": [ "Q₈ irrep a" ], "searchTerms": [ "q8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of Q₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled a of Q₈, specified by its character on the named conjugacy classes. i maps to +1 and j to -1", "definitionEvidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "a", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:q-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:q-8", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "a", "displayTex": "Q_8:\\quad a", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "−1", "latex": "-1", "labelTeX": "-1", "size": "1", "order": 2, "representative": "-1" }, { "id": "4i", "label": "i", "latex": "i", "labelTeX": "i", "size": "2", "order": 4, "representative": "i" }, { "id": "4j", "label": "j", "latex": "j", "labelTeX": "j", "size": "2", "order": 4, "representative": "j" }, { "id": "4k", "label": "k", "latex": "k", "labelTeX": "k", "size": "2", "order": 4, "representative": "k" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "q8", "operation": "tensor", "left": "a", "right": "a", "copies": "1", "degree": "2" } } } }, { "id": "representation:q-8-b", "slug": "q-8-b", "name": "Q₈ irreducible representation b", "symbol": "Q₈ · b", "category": "finite", "level": "executable", "aliases": [ "Q₈ irrep b" ], "searchTerms": [ "q8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of Q₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled b of Q₈, specified by its character on the named conjugacy classes. i maps to -1 and j to +1", "definitionEvidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "b", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:q-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:q-8", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "b", "displayTex": "Q_8:\\quad b", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "−1", "latex": "-1", "labelTeX": "-1", "size": "1", "order": 2, "representative": "-1" }, { "id": "4i", "label": "i", "latex": "i", "labelTeX": "i", "size": "2", "order": 4, "representative": "i" }, { "id": "4j", "label": "j", "latex": "j", "labelTeX": "j", "size": "2", "order": 4, "representative": "j" }, { "id": "4k", "label": "k", "latex": "k", "labelTeX": "k", "size": "2", "order": 4, "representative": "k" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "q8", "operation": "tensor", "left": "b", "right": "b", "copies": "1", "degree": "2" } } } }, { "id": "representation:q-8-c", "slug": "q-8-c", "name": "Q₈ irreducible representation c", "symbol": "Q₈ · c", "category": "finite", "level": "executable", "aliases": [ "Q₈ irrep c" ], "searchTerms": [ "q8", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of Q₈, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled c of Q₈, specified by its character on the named conjugacy classes. i maps to -1 and j to -1", "definitionEvidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "c", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:q-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:q-8", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "c", "displayTex": "Q_8:\\quad c", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "−1", "latex": "-1", "labelTeX": "-1", "size": "1", "order": 2, "representative": "-1" }, { "id": "4i", "label": "i", "latex": "i", "labelTeX": "i", "size": "2", "order": 4, "representative": "i" }, { "id": "4j", "label": "j", "latex": "j", "labelTeX": "j", "size": "2", "order": 4, "representative": "j" }, { "id": "4k", "label": "k", "latex": "k", "labelTeX": "k", "size": "2", "order": 4, "representative": "k" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "q8", "operation": "tensor", "left": "c", "right": "c", "copies": "1", "degree": "2" } } } }, { "id": "representation:q-8-2", "slug": "q-8-2", "name": "Q₈ irreducible representation 2", "symbol": "Q₈ · 2", "category": "finite", "level": "executable", "aliases": [ "Q₈ irrep 2" ], "searchTerms": [ "q8", "Quaternionic", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 2-dimensional ordinary complex irrep of Q₈, of quaternionic type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 2 of Q₈, specified by its character on the named conjugacy classes. Faithful quaternionic-type doublet; its character is real-valued", "definitionEvidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Quaternionic", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "-1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "2", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; quaternionic describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:q-8", "label": "Representation domain", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:q-8", "category": "ordinary-complex", "field": "C", "dimension": 2, "labels": "2", "displayTex": "Q_8:\\quad \\mathbf{2}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "1" }, { "id": "2z", "label": "−1", "latex": "-1", "labelTeX": "-1", "size": "1", "order": 2, "representative": "-1" }, { "id": "4i", "label": "i", "latex": "i", "labelTeX": "i", "size": "2", "order": 4, "representative": "i" }, { "id": "4j", "label": "j", "latex": "j", "labelTeX": "j", "size": "2", "order": 4, "representative": "j" }, { "id": "4k", "label": "k", "latex": "k", "labelTeX": "k", "size": "2", "order": 4, "representative": "k" } ], "values": [ { "text": "2", "latex": "2", "coefficients": [ "2", "0" ], "basis": "1" }, { "text": "-2", "latex": "-2", "coefficients": [ "-2", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "q8", "operation": "tensor", "left": "2", "right": "2", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-4-1", "slug": "a-4-1", "name": "A₄ irreducible representation 1", "symbol": "A₄ · 1", "category": "finite", "level": "executable", "aliases": [ "A₄ irrep 1" ], "searchTerms": [ "a4", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of A₄, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1 of A₄, specified by its character on the named conjugacy classes. Trivial representation", "definitionEvidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123).", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-4", "label": "Representation domain", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-4", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1", "displayTex": "A_4:\\quad \\mathbf{1}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "4", "order": 3, "representative": "(123)" }, { "id": "3b", "label": "(132)", "latex": "(132)", "labelTeX": "(132)", "size": "4", "order": 3, "representative": "(132)" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" } ] }, "capabilityPresets": { "finite-characters": { "group": "a4", "operation": "tensor", "left": "1", "right": "1", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-4-1p", "slug": "a-4-1p", "name": "A₄ irreducible representation 1′", "symbol": "A₄ · 1′", "category": "finite", "level": "executable", "aliases": [ "A₄ irrep 1′" ], "searchTerms": [ "a4", "Complex", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of A₄, of complex type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1′ of A₄, specified by its character on the named conjugacy classes. Nontrivial quotient character with value omega on (123)", "definitionEvidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Complex", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "0", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1″", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123).", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; complex describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-4", "label": "Representation domain", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] }, { "type": "conjugate", "target": "representation:a-4-1pp", "label": "Complex conjugate representation", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-4", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1p", "displayTex": "A_4:\\quad \\mathbf{1}'", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "4", "order": 3, "representative": "(123)" }, { "id": "3b", "label": "(132)", "latex": "(132)", "labelTeX": "(132)", "size": "4", "order": 3, "representative": "(132)" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "ω", "latex": "\\omega", "coefficients": [ "0", "1" ], "basis": "1,omega" }, { "text": "ω²", "latex": "\\omega^2", "coefficients": [ "-1", "-1" ], "basis": "1,omega" } ] }, "capabilityPresets": { "finite-characters": { "group": "a4", "operation": "tensor", "left": "1p", "right": "1pp", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-4-1pp", "slug": "a-4-1pp", "name": "A₄ irreducible representation 1″", "symbol": "A₄ · 1″", "category": "finite", "level": "executable", "aliases": [ "A₄ irrep 1″" ], "searchTerms": [ "a4", "Complex", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of A₄, of complex type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1″ of A₄, specified by its character on the named conjugacy classes. Complex conjugate of 1′", "definitionEvidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Complex", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "0", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1′", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123).", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; complex describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-4", "label": "Representation domain", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] }, { "type": "conjugate", "target": "representation:a-4-1p", "label": "Complex conjugate representation", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-4", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1pp", "displayTex": "A_4:\\quad \\mathbf{1}''", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "4", "order": 3, "representative": "(123)" }, { "id": "3b", "label": "(132)", "latex": "(132)", "labelTeX": "(132)", "size": "4", "order": 3, "representative": "(132)" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,omega" }, { "text": "ω²", "latex": "\\omega^2", "coefficients": [ "-1", "-1" ], "basis": "1,omega" }, { "text": "ω", "latex": "\\omega", "coefficients": [ "0", "1" ], "basis": "1,omega" } ] }, "capabilityPresets": { "finite-characters": { "group": "a4", "operation": "tensor", "left": "1pp", "right": "1p", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-4-3", "slug": "a-4-3", "name": "A₄ irreducible representation 3", "symbol": "A₄ · 3", "category": "finite", "level": "executable", "aliases": [ "A₄ irrep 3" ], "searchTerms": [ "a4", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 3-dimensional ordinary complex irrep of A₄, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 3 of A₄, specified by its character on the named conjugacy classes. Faithful real-type rotational triplet", "definitionEvidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "3", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123).", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-4", "label": "Representation domain", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-4", "category": "ordinary-complex", "field": "C", "dimension": 3, "labels": "3", "displayTex": "A_4:\\quad \\mathbf{3}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "4", "order": 3, "representative": "(123)" }, { "id": "3b", "label": "(132)", "latex": "(132)", "labelTeX": "(132)", "size": "4", "order": 3, "representative": "(132)" } ], "values": [ { "text": "3", "latex": "3", "coefficients": [ "3", "0" ], "basis": "1,omega" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1,omega" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1,omega" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1,omega" } ] }, "capabilityPresets": { "finite-characters": { "group": "a4", "operation": "tensor", "left": "3", "right": "3", "copies": "1", "degree": "2" } } } }, { "id": "representation:s-4-1", "slug": "s-4-1", "name": "S₄ irreducible representation 1", "symbol": "S₄ · 1", "category": "finite", "level": "executable", "aliases": [ "S₄ irrep 1" ], "searchTerms": [ "s4", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of S₄, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1 of S₄, specified by its character on the named conjugacy classes. Trivial representation", "definitionEvidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:s-4", "label": "Representation domain", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:s-4", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1", "displayTex": "S_4:\\quad \\mathbf{1}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)", "latex": "(12)", "labelTeX": "(12)", "size": "6", "order": 2, "representative": "(12)" }, { "id": "2b", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "8", "order": 3, "representative": "(123)" }, { "id": "4a", "label": "(1234)", "latex": "(1234)", "labelTeX": "(1234)", "size": "6", "order": 4, "representative": "(1234)" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "s4", "operation": "tensor", "left": "1", "right": "1", "copies": "1", "degree": "2" } } } }, { "id": "representation:s-4-sgn", "slug": "s-4-sgn", "name": "S₄ irreducible representation sgn", "symbol": "S₄ · sgn", "category": "finite", "level": "executable", "aliases": [ "S₄ irrep sgn" ], "searchTerms": [ "s4", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of S₄, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled sgn of S₄, specified by its character on the named conjugacy classes. Sign of a permutation", "definitionEvidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "sgn", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:s-4", "label": "Representation domain", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:s-4", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "sgn", "displayTex": "S_4:\\quad \\mathrm{sgn}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)", "latex": "(12)", "labelTeX": "(12)", "size": "6", "order": 2, "representative": "(12)" }, { "id": "2b", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "8", "order": 3, "representative": "(123)" }, { "id": "4a", "label": "(1234)", "latex": "(1234)", "labelTeX": "(1234)", "size": "6", "order": 4, "representative": "(1234)" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "s4", "operation": "tensor", "left": "sgn", "right": "sgn", "copies": "1", "degree": "2" } } } }, { "id": "representation:s-4-2", "slug": "s-4-2", "name": "S₄ irreducible representation 2", "symbol": "S₄ · 2", "category": "finite", "level": "executable", "aliases": [ "S₄ irrep 2" ], "searchTerms": [ "s4", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 2-dimensional ordinary complex irrep of S₄, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 2 of S₄, specified by its character on the named conjugacy classes. Standard S3 doublet pulled back through S4 → S3", "definitionEvidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "2", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:s-4", "label": "Representation domain", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:s-4", "category": "ordinary-complex", "field": "C", "dimension": 2, "labels": "2", "displayTex": "S_4:\\quad \\mathbf{2}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)", "latex": "(12)", "labelTeX": "(12)", "size": "6", "order": 2, "representative": "(12)" }, { "id": "2b", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "8", "order": 3, "representative": "(123)" }, { "id": "4a", "label": "(1234)", "latex": "(1234)", "labelTeX": "(1234)", "size": "6", "order": 4, "representative": "(1234)" } ], "values": [ { "text": "2", "latex": "2", "coefficients": [ "2", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" }, { "text": "2", "latex": "2", "coefficients": [ "2", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "s4", "operation": "tensor", "left": "2", "right": "2", "copies": "1", "degree": "2" } } } }, { "id": "representation:s-4-3", "slug": "s-4-3", "name": "S₄ irreducible representation 3", "symbol": "S₄ · 3", "category": "finite", "level": "executable", "aliases": [ "S₄ irrep 3" ], "searchTerms": [ "s4", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 3-dimensional ordinary complex irrep of S₄, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 3 of S₄, specified by its character on the named conjugacy classes. Standard triplet; the determinant is sgn", "definitionEvidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "3", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:s-4", "label": "Representation domain", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:s-4", "category": "ordinary-complex", "field": "C", "dimension": 3, "labels": "3", "displayTex": "S_4:\\quad \\mathbf{3}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)", "latex": "(12)", "labelTeX": "(12)", "size": "6", "order": 2, "representative": "(12)" }, { "id": "2b", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "8", "order": 3, "representative": "(123)" }, { "id": "4a", "label": "(1234)", "latex": "(1234)", "labelTeX": "(1234)", "size": "6", "order": 4, "representative": "(1234)" } ], "values": [ { "text": "3", "latex": "3", "coefficients": [ "3", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "s4", "operation": "tensor", "left": "3", "right": "3", "copies": "1", "degree": "2" } } } }, { "id": "representation:s-4-3p", "slug": "s-4-3p", "name": "S₄ irreducible representation 3′", "symbol": "S₄ · 3′", "category": "finite", "level": "executable", "aliases": [ "S₄ irrep 3′" ], "searchTerms": [ "s4", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 3-dimensional ordinary complex irrep of S₄, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 3′ of S₄, specified by its character on the named conjugacy classes. Sign twist of the standard triplet; the cubic-rotation representation", "definitionEvidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "3′", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:s-4", "label": "Representation domain", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:s-4", "category": "ordinary-complex", "field": "C", "dimension": 3, "labels": "3p", "displayTex": "S_4:\\quad \\mathbf{3}'", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)", "latex": "(12)", "labelTeX": "(12)", "size": "6", "order": 2, "representative": "(12)" }, { "id": "2b", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "3", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "8", "order": 3, "representative": "(123)" }, { "id": "4a", "label": "(1234)", "latex": "(1234)", "labelTeX": "(1234)", "size": "6", "order": 4, "representative": "(1234)" } ], "values": [ { "text": "3", "latex": "3", "coefficients": [ "3", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1" } ] }, "capabilityPresets": { "finite-characters": { "group": "s4", "operation": "tensor", "left": "3p", "right": "3p", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-5-1", "slug": "a-5-1", "name": "A₅ irreducible representation 1", "symbol": "A₅ · 1", "category": "finite", "level": "executable", "aliases": [ "A₅ irrep 1" ], "searchTerms": [ "a5", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 1-dimensional ordinary complex irrep of A₅, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 1 of A₅, specified by its character on the named conjugacy classes. Trivial representation", "definitionEvidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-5", "label": "Representation domain", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-5", "category": "ordinary-complex", "field": "C", "dimension": 1, "labels": "1", "displayTex": "A_5:\\quad \\mathbf{1}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "15", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "20", "order": 3, "representative": "(123)" }, { "id": "5a", "label": "c", "latex": "c", "labelTeX": "c", "size": "12", "order": 5, "representative": "(12345)" }, { "id": "5b", "label": "c²", "latex": "c^2", "labelTeX": "c^2", "size": "12", "order": 5, "representative": "(13524)" } ], "values": [ { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,phi" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,phi" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,phi" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,phi" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,phi" } ] }, "capabilityPresets": { "finite-characters": { "group": "a5", "operation": "tensor", "left": "1", "right": "1", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-5-3", "slug": "a-5-3", "name": "A₅ irreducible representation 3", "symbol": "A₅ · 3", "category": "finite", "level": "executable", "aliases": [ "A₅ irrep 3" ], "searchTerms": [ "a5", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 3-dimensional ordinary complex irrep of A₅, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 3 of A₅, specified by its character on the named conjugacy classes. Real-type triplet with phi on the chosen five-cycle c", "definitionEvidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "3", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-5", "label": "Representation domain", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-5", "category": "ordinary-complex", "field": "C", "dimension": 3, "labels": "3", "displayTex": "A_5:\\quad \\mathbf{3}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "15", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "20", "order": 3, "representative": "(123)" }, { "id": "5a", "label": "c", "latex": "c", "labelTeX": "c", "size": "12", "order": 5, "representative": "(12345)" }, { "id": "5b", "label": "c²", "latex": "c^2", "labelTeX": "c^2", "size": "12", "order": 5, "representative": "(13524)" } ], "values": [ { "text": "3", "latex": "3", "coefficients": [ "3", "0" ], "basis": "1,phi" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1,phi" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1,phi" }, { "text": "φ", "latex": "\\varphi", "coefficients": [ "0", "1" ], "basis": "1,phi" }, { "text": "1-φ", "latex": "1-\\varphi", "coefficients": [ "1", "-1" ], "basis": "1,phi" } ] }, "capabilityPresets": { "finite-characters": { "group": "a5", "operation": "tensor", "left": "3", "right": "3", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-5-3p", "slug": "a-5-3p", "name": "A₅ irreducible representation 3′", "symbol": "A₅ · 3′", "category": "finite", "level": "executable", "aliases": [ "A₅ irrep 3′" ], "searchTerms": [ "a5", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 3-dimensional ordinary complex irrep of A₅, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 3′ of A₅, specified by its character on the named conjugacy classes. Inequivalent real-type triplet obtained by exchanging five-cycle classes", "definitionEvidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "3′", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-5", "label": "Representation domain", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-5", "category": "ordinary-complex", "field": "C", "dimension": 3, "labels": "3p", "displayTex": "A_5:\\quad \\mathbf{3}'", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "15", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "20", "order": 3, "representative": "(123)" }, { "id": "5a", "label": "c", "latex": "c", "labelTeX": "c", "size": "12", "order": 5, "representative": "(12345)" }, { "id": "5b", "label": "c²", "latex": "c^2", "labelTeX": "c^2", "size": "12", "order": 5, "representative": "(13524)" } ], "values": [ { "text": "3", "latex": "3", "coefficients": [ "3", "0" ], "basis": "1,phi" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1,phi" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1,phi" }, { "text": "1-φ", "latex": "1-\\varphi", "coefficients": [ "1", "-1" ], "basis": "1,phi" }, { "text": "φ", "latex": "\\varphi", "coefficients": [ "0", "1" ], "basis": "1,phi" } ] }, "capabilityPresets": { "finite-characters": { "group": "a5", "operation": "tensor", "left": "3p", "right": "3p", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-5-4", "slug": "a-5-4", "name": "A₅ irreducible representation 4", "symbol": "A₅ · 4", "category": "finite", "level": "executable", "aliases": [ "A₅ irrep 4" ], "searchTerms": [ "a5", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 4-dimensional ordinary complex irrep of A₅, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 4 of A₅, specified by its character on the named conjugacy classes. Five-letter permutation representation minus the invariant line", "definitionEvidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "4", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "4", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-5", "label": "Representation domain", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-5", "category": "ordinary-complex", "field": "C", "dimension": 4, "labels": "4", "displayTex": "A_5:\\quad \\mathbf{4}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "15", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "20", "order": 3, "representative": "(123)" }, { "id": "5a", "label": "c", "latex": "c", "labelTeX": "c", "size": "12", "order": 5, "representative": "(12345)" }, { "id": "5b", "label": "c²", "latex": "c^2", "labelTeX": "c^2", "size": "12", "order": 5, "representative": "(13524)" } ], "values": [ { "text": "4", "latex": "4", "coefficients": [ "4", "0" ], "basis": "1,phi" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1,phi" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,phi" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1,phi" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1,phi" } ] }, "capabilityPresets": { "finite-characters": { "group": "a5", "operation": "tensor", "left": "4", "right": "4", "copies": "1", "degree": "2" } } } }, { "id": "representation:a-5-5", "slug": "a-5-5", "name": "A₅ irreducible representation 5", "symbol": "A₅ · 5", "category": "finite", "level": "executable", "aliases": [ "A₅ irrep 5" ], "searchTerms": [ "a5", "Real", "character", "tensor product" ], "tags": [ "Ordinary complex", "Character", "Irreducible" ], "summary": "A 5-dimensional ordinary complex irrep of A₅, of real type, with an exact character and executable tensor operations.", "definition": "The ordinary complex irreducible representation labeled 5 of A₅, specified by its character on the named conjugacy classes. Real-type five-dimensional irrep", "definitionEvidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] }, { "key": "conjugate", "label": "Complex conjugate irrep", "value": "5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Character and identity", "paragraphs": [ { "text": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Tensor operations and physical use", "paragraphs": [ { "text": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Conventions and realization", "paragraphs": [ { "text": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:a-5", "label": "Representation domain", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ] } ], "capabilities": [ "finite-characters" ], "kind": "representation", "data": { "domain": "group:a-5", "category": "ordinary-complex", "field": "C", "dimension": 5, "labels": "5", "displayTex": "A_5:\\quad \\mathbf{5}", "conventionId": "finite-ordinary-complex-v1", "character": { "classes": [ { "id": "1", "label": "1", "latex": "1", "labelTeX": "1", "size": "1", "order": 1, "representative": "()" }, { "id": "2a", "label": "(12)(34)", "latex": "(12)(34)", "labelTeX": "(12)(34)", "size": "15", "order": 2, "representative": "(12)(34)" }, { "id": "3a", "label": "(123)", "latex": "(123)", "labelTeX": "(123)", "size": "20", "order": 3, "representative": "(123)" }, { "id": "5a", "label": "c", "latex": "c", "labelTeX": "c", "size": "12", "order": 5, "representative": "(12345)" }, { "id": "5b", "label": "c²", "latex": "c^2", "labelTeX": "c^2", "size": "12", "order": 5, "representative": "(13524)" } ], "values": [ { "text": "5", "latex": "5", "coefficients": [ "5", "0" ], "basis": "1,phi" }, { "text": "1", "latex": "1", "coefficients": [ "1", "0" ], "basis": "1,phi" }, { "text": "-1", "latex": "-1", "coefficients": [ "-1", "0" ], "basis": "1,phi" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1,phi" }, { "text": "0", "latex": "0", "coefficients": [ "0", "0" ], "basis": "1,phi" } ] }, "capabilityPresets": { "finite-characters": { "group": "a5", "operation": "tensor", "left": "5", "right": "5", "copies": "1", "degree": "2" } } } }, { "id": "representation:su-2-weight-1", "slug": "su-2-weight-1", "name": "Doublet of SU(2)", "symbol": "A1 · (1)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "SU(2)", "Doublet", "2-dimensional", "Quaternionic" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (1) of the specified simply connected compact group: complex dimension 2, quaternionic type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of SU(2) with Dynkin labels (1) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "3/4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "1/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Quaternionic", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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The test for descent to its adjoint global form gives yes. 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "A", "rank": "4", "labels": "1,0,0,1" }, "sun-tensor-products": { "title": "Adjoint of SU(5) times its dual", "n": "5", "left": "1,0,0,1", "right": "1,0,0,1" }, "charged-branching": { "title": "Block restriction of Adjoint of SU(5)", "n": "5", "p": "3", "labels": "1,0,0,1" } } } }, { "id": "representation:sp-2-weight-1-0", "slug": "sp-2-weight-1-0", "name": "Defining representation of Sp(2) = USp(4)", "symbol": "C2 · (1,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Sp(2) = USp(4)", "Defining representation", "4-dimensional", "Quaternionic" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (1,0) of the specified simply connected compact group: complex dimension 4, quaternionic type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of Sp(2) = USp(4) with Dynkin labels (1,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; 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tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "1/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Quaternionic", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; 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Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "Nodes 1,…,r run along the chain; node r is long. 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "C", "rank": "2", "labels": "1,0" } } } }, { "id": "representation:sp-2-weight-0-1", "slug": "sp-2-weight-0-1", "name": "Five-dimensional irrep of Sp(2) = USp(4)", "symbol": "C2 · (0,1)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Sp(2) = USp(4)", "Five-dimensional irrep", "5-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,1) of the specified simply connected compact group: complex dimension 5, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of Sp(2) = USp(4) with Dynkin labels (0,1) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "Nodes 1,…,r run along the chain; 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "D", "rank": "4", "labels": "0,0,0,1" } } } }, { "id": "representation:spin-10-weight-1-0-0-0-0", "slug": "spin-10-weight-1-0-0-0-0", "name": "Vector of Spin(10)", "symbol": "D5 · (1,0,0,0,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Spin(10)", "Vector", "10-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (1,0,0,0,0) of the specified simply connected compact group: complex dimension 10, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of Spin(10) with Dynkin labels (1,0,0,0,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "D", "rank": "5", "labels": "1,0,0,0,0" } } } }, { "id": "representation:spin-10-weight-0-0-0-0-1", "slug": "spin-10-weight-0-0-0-0-1", "name": "Chiral spinor of Spin(10)", "symbol": "D5 · (0,0,0,0,1)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Spin(10)", "Chiral spinor", "16-dimensional", "Complex" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,0,0,0,1) of the specified simply connected compact group: complex dimension 16, complex type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of Spin(10) with Dynkin labels (0,0,0,0,1) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "D", "rank": "5", "labels": "0,0,0,0,1" } } } }, { "id": "representation:spin-10-weight-0-0-0-1-0", "slug": "spin-10-weight-0-0-0-1-0", "name": "Conjugate chiral spinor of Spin(10)", "symbol": "D5 · (0,0,0,1,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Spin(10)", "Conjugate chiral spinor", "16-dimensional", "Complex" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,0,0,1,0) of the specified simply connected compact group: complex dimension 16, complex type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of Spin(10) with Dynkin labels (0,0,0,1,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "The chain 1,…,r−2 branches to nodes r−1 and r. 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "D", "rank": "5", "labels": "0,0,0,1,0" } } } }, { "id": "representation:spin-10-weight-0-1-0-0-0", "slug": "spin-10-weight-0-1-0-0-0", "name": "Adjoint of Spin(10)", "symbol": "D5 · (0,1,0,0,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Spin(10)", "Adjoint", "45-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,1,0,0,0) of the specified simply connected compact group: complex dimension 45, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of Spin(10) with Dynkin labels (0,1,0,0,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "45", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. 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A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "D", "rank": "5", "labels": "0,1,0,0,0" } } } }, { "id": "representation:g-2-compact-weight-1-0", "slug": "g-2-compact-weight-1-0", "name": "Fundamental of G2 (compact)", "symbol": "G2 · (1,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "G2 (compact)", "Fundamental", "7-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (1,0) of the specified simply connected compact group: complex dimension 7, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of G2 (compact) with Dynkin labels (1,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "7", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "Node 1 is short and node 2 long. Weights 1 and 2 have dimensions 7 and 14, respectively. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. 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A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "G", "rank": "2", "labels": "1,0" } } } }, { "id": "representation:g-2-compact-weight-0-1", "slug": "g-2-compact-weight-0-1", "name": "Adjoint of G2 (compact)", "symbol": "G2 · (0,1)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "G2 (compact)", "Adjoint", "14-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,1) of the specified simply connected compact group: complex dimension 14, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of G2 (compact) with Dynkin labels (0,1) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; 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documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "Node 1 is short and node 2 long. Weights 1 and 2 have dimensions 7 and 14, respectively. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Use in a calculation", "paragraphs": [ { "text": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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Weights 1 and 2 have dimensions 7 and 14, respectively.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "labels": [ 0, 1 ], "dimension": "14", "casimir": { "numerator": "4", "denominator": "1", "text": "4", "latex": "4" }, "index": { "numerator": "4", "denominator": "1", "text": "4", "latex": "4" }, "conjugate": [ 0, 1 ], "selfDual": true, "reality": "Real", "frobeniusSchurIndicator": 1, "adjointDescent": true, "centerCharacterOrder": "1", "centerKernelOrder": "1", "centerCharge": null, "centerChargeModulus": null, "rootBasisCoordinates": [ { "numerator": "3", "denominator": "1", "text": "3", "latex": "3" }, { "numerator": "2", "denominator": "1", "text": "2", "latex": "2" } ], "positiveCoroots": [ [ 0, 1 ], [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 3 ] ], "factors": [ { "coroot": [ 0, 1 ], "numerator": "2", "denominator": "1" }, { "coroot": [ 1, 0 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 1, 1 ], "numerator": "3", "denominator": "2" }, { "coroot": [ 1, 2 ], "numerator": "5", "denominator": "3" }, { "coroot": [ 1, 3 ], "numerator": "7", "denominator": "4" }, { "coroot": [ 2, 3 ], "numerator": "8", "denominator": "5" } ], "algorithmVersion": "lie-representations-1.0.0", "category": "Finite-dimensional continuous complex irreducible representations of the compact simply connected group", "method": "Exact Weyl product over positive coroots; rational weight-metric Casimir; root-lattice descent; central-character Frobenius–Schur indicator.", "checks": { "positiveCorootCount": true, "dimensionIntegral": true, "weightGramSymmetric": true, "centralCharacterOrderDividesCenter": true }, "sources": [ { "id": "workbench-etingof-lie", "title": "Lie Groups and Lie Algebras I", "authors": [ "Pavel Etingof" ], "url": "https://math.mit.edu/~etingof/lnlg.pdf", "kind": "lecture-notes", "locatorHint": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-g", "title": "SageMath: Root system data for type G", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_G.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "G", "rank": "2", "labels": "0,1" } } } }, { "id": "representation:f-4-compact-weight-0-0-0-1", "slug": "f-4-compact-weight-0-0-0-1", "name": "Minimal representation of F4 (compact)", "symbol": "F4 · (0,0,0,1)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "F4 (compact)", "Minimal representation", "26-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,0,0,1) of the specified simply connected compact group: complex dimension 26, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of F4 (compact) with Dynkin labels (0,0,0,1) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "6", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "The chain is 1–2–3–4; nodes 1 and 2 are long, nodes 3 and 4 short. Weight 4 has dimension 26. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Use in a calculation", "paragraphs": [ { "text": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-f", "title": "SageMath: Root system data for type F", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_F.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "F", "rank": "4", "labels": "0,0,0,1" } } } }, { "id": "representation:f-4-compact-weight-1-0-0-0", "slug": "f-4-compact-weight-1-0-0-0", "name": "Adjoint of F4 (compact)", "symbol": "F4 · (1,0,0,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "F4 (compact)", "Adjoint", "52-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (1,0,0,0) of the specified simply connected compact group: complex dimension 52, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of F4 (compact) with Dynkin labels (1,0,0,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "52", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "9", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "9", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "The chain is 1–2–3–4; nodes 1 and 2 are long, nodes 3 and 4 short. Weight 4 has dimension 26. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Use in a calculation", "paragraphs": [ { "text": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:f-4-compact", "label": "Representation domain", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ] } ], "capabilities": [ "lie-representations" ], "kind": "representation", "data": { "domain": "group:f-4-compact", "category": "finite-dimensional-continuous", "field": "C", "dimension": 52, "labels": "(1,0,0,0)", "displayTex": "F_{4}:\\quad V_{(1,0,0,0)}", "dynkinLabels": [ 1, 0, 0, 0 ], "conventionId": "lie-bourbaki-long-root-2-v1", "calculation": { "key": "F4", "type": "F", "rank": 4, "cartanMatrix": [ [ 2, -1, 0, 0 ], [ -1, 2, -1, 0 ], [ 0, -2, 2, -1 ], [ 0, 0, -1, 2 ] ], "rootLengthsSquared": [ "2", "2", "1", "1" ], "positiveCorootCount": 24, "algebraDimension": 52, "simplyConnectedGroup": "F4 (compact)", "center": "1", "centerOrder": 1, "dualCoxeterNumber": 9, "adjointLabels": [ 1, 0, 0, 0 ], "exampleLabels": [ 0, 0, 0, 1 ], "numbering": "The chain is 1–2–3–4; nodes 1 and 2 are long, nodes 3 and 4 short. Weight 4 has dimension 26.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "labels": [ 1, 0, 0, 0 ], "dimension": "52", "casimir": { "numerator": "9", "denominator": "1", "text": "9", "latex": "9" }, "index": { "numerator": "9", "denominator": "1", "text": "9", "latex": "9" }, "conjugate": [ 1, 0, 0, 0 ], "selfDual": true, "reality": "Real", "frobeniusSchurIndicator": 1, "adjointDescent": true, "centerCharacterOrder": "1", "centerKernelOrder": "1", "centerCharge": null, "centerChargeModulus": null, "rootBasisCoordinates": [ { "numerator": "2", "denominator": "1", "text": "2", "latex": "2" }, { "numerator": "3", "denominator": "1", "text": "3", "latex": "3" }, { "numerator": "4", "denominator": "1", "text": "4", "latex": "4" }, { "numerator": "2", "denominator": "1", "text": "2", "latex": "2" } ], "positiveCoroots": [ [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 1, 1, 0 ], [ 1, 1, 0, 0 ], [ 0, 1, 1, 1 ], [ 0, 2, 1, 0 ], [ 1, 1, 1, 0 ], [ 0, 2, 1, 1 ], [ 1, 1, 1, 1 ], [ 1, 2, 1, 0 ], [ 0, 2, 2, 1 ], [ 1, 2, 1, 1 ], [ 2, 2, 1, 0 ], [ 1, 2, 2, 1 ], [ 2, 2, 1, 1 ], [ 1, 3, 2, 1 ], [ 2, 2, 2, 1 ], [ 2, 3, 2, 1 ], [ 2, 4, 2, 1 ], [ 2, 4, 3, 1 ], [ 2, 4, 3, 2 ] ], "factors": [ { "coroot": [ 0, 0, 0, 1 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 0, 0, 1, 0 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 0, 1, 0, 0 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 1, 0, 0, 0 ], "numerator": "2", "denominator": "1" }, { "coroot": [ 0, 0, 1, 1 ], "numerator": "2", "denominator": "2" }, { "coroot": [ 0, 1, 1, 0 ], "numerator": "2", "denominator": "2" }, { "coroot": [ 1, 1, 0, 0 ], "numerator": "3", "denominator": "2" }, { "coroot": [ 0, 1, 1, 1 ], "numerator": "3", "denominator": "3" }, { "coroot": [ 0, 2, 1, 0 ], "numerator": "3", "denominator": "3" }, { "coroot": [ 1, 1, 1, 0 ], "numerator": "4", "denominator": "3" }, { "coroot": [ 0, 2, 1, 1 ], "numerator": "4", "denominator": "4" }, { "coroot": [ 1, 1, 1, 1 ], "numerator": "5", "denominator": "4" }, { "coroot": [ 1, 2, 1, 0 ], "numerator": "5", "denominator": "4" }, { "coroot": [ 0, 2, 2, 1 ], "numerator": "5", "denominator": "5" }, { "coroot": [ 1, 2, 1, 1 ], "numerator": "6", "denominator": "5" }, { "coroot": [ 2, 2, 1, 0 ], "numerator": "7", "denominator": "5" }, { "coroot": [ 1, 2, 2, 1 ], "numerator": "7", "denominator": "6" }, { "coroot": [ 2, 2, 1, 1 ], "numerator": "8", "denominator": "6" }, { "coroot": [ 1, 3, 2, 1 ], "numerator": "8", "denominator": "7" }, { "coroot": [ 2, 2, 2, 1 ], "numerator": "9", "denominator": "7" }, { "coroot": [ 2, 3, 2, 1 ], "numerator": "10", "denominator": "8" }, { "coroot": [ 2, 4, 2, 1 ], "numerator": "11", "denominator": "9" }, { "coroot": [ 2, 4, 3, 1 ], "numerator": "12", "denominator": "10" }, { "coroot": [ 2, 4, 3, 2 ], "numerator": "13", "denominator": "11" } ], "algorithmVersion": "lie-representations-1.0.0", "category": "Finite-dimensional continuous complex irreducible representations of the compact simply connected group", "method": "Exact Weyl product over positive coroots; rational weight-metric Casimir; root-lattice descent; central-character Frobenius–Schur indicator.", "checks": { "positiveCorootCount": true, "dimensionIntegral": true, "weightGramSymmetric": true, "centralCharacterOrderDividesCenter": true }, "sources": [ { "id": "workbench-etingof-lie", "title": "Lie Groups and Lie Algebras I", "authors": [ "Pavel Etingof" ], "url": "https://math.mit.edu/~etingof/lnlg.pdf", "kind": "lecture-notes", "locatorHint": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-f", "title": "SageMath: Root system data for type F", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_F.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "F", "rank": "4", "labels": "1,0,0,0" } } } }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "slug": "e-6-compact-sc-weight-1-0-0-0-0-0", "name": "27-dimensional representation of E6 (compact simply connected)", "symbol": "E6 · (1,0,0,0,0,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "E6 (compact simply connected)", "27-dimensional representation", "27-dimensional", "Complex" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (1,0,0,0,0,0) of the specified simply connected compact group: complex dimension 27, complex type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of E6 (compact simply connected) with Dynkin labels (1,0,0,0,0,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "27", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "26/3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Use in a calculation", "paragraphs": [ { "text": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:e-6-compact-sc", "label": "Representation domain", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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node 2 attaches to node 4.", "conventionId": "lie-bourbaki-long-root-2-v1", "cartanConvention": "A_ij = ", "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.", "labels": [ 1, 0, 0, 0, 0, 0 ], "dimension": "27", "casimir": { "numerator": "26", "denominator": "3", "text": "26/3", "latex": "\\frac{26}{3}" }, "index": { "numerator": "3", "denominator": "1", "text": "3", "latex": "3" }, "conjugate": [ 0, 0, 0, 0, 0, 1 ], "selfDual": false, "reality": "Complex", "frobeniusSchurIndicator": 0, "adjointDescent": false, "centerCharacterOrder": "3", "centerKernelOrder": "1", "centerCharge": null, "centerChargeModulus": null, "rootBasisCoordinates": [ { "numerator": "4", "denominator": "3", "text": "4/3", "latex": "\\frac{4}{3}" }, { "numerator": "1", "denominator": "1", "text": "1", "latex": "1" }, { "numerator": "5", "denominator": "3", "text": "5/3", "latex": "\\frac{5}{3}" }, { "numerator": "2", "denominator": "1", "text": "2", "latex": "2" }, { "numerator": "4", "denominator": "3", "text": "4/3", "latex": "\\frac{4}{3}" }, { "numerator": "2", "denominator": "3", "text": "2/3", "latex": "\\frac{2}{3}" } ], "positiveCoroots": [ [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 1 ], [ 0, 0, 0, 1, 1, 0 ], [ 0, 0, 1, 1, 0, 0 ], [ 0, 1, 0, 1, 0, 0 ], [ 1, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 1, 1 ], [ 0, 0, 1, 1, 1, 0 ], [ 0, 1, 0, 1, 1, 0 ], [ 0, 1, 1, 1, 0, 0 ], [ 1, 0, 1, 1, 0, 0 ], [ 0, 0, 1, 1, 1, 1 ], [ 0, 1, 0, 1, 1, 1 ], [ 0, 1, 1, 1, 1, 0 ], [ 1, 0, 1, 1, 1, 0 ], [ 1, 1, 1, 1, 0, 0 ], [ 0, 1, 1, 1, 1, 1 ], [ 0, 1, 1, 2, 1, 0 ], [ 1, 0, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 0 ], [ 0, 1, 1, 2, 1, 1 ], [ 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 2, 1, 0 ], [ 0, 1, 1, 2, 2, 1 ], [ 1, 1, 1, 2, 1, 1 ], [ 1, 1, 2, 2, 1, 0 ], [ 1, 1, 1, 2, 2, 1 ], [ 1, 1, 2, 2, 1, 1 ], [ 1, 1, 2, 2, 2, 1 ], [ 1, 1, 2, 3, 2, 1 ], [ 1, 2, 2, 3, 2, 1 ] ], "factors": [ { "coroot": [ 0, 0, 0, 0, 0, 1 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 0, 0, 0, 0, 1, 0 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 0, 0, 0, 1, 0, 0 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 0, 0, 1, 0, 0, 0 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 0, 1, 0, 0, 0, 0 ], "numerator": "1", "denominator": "1" }, { "coroot": [ 1, 0, 0, 0, 0, 0 ], "numerator": "2", "denominator": "1" }, { "coroot": [ 0, 0, 0, 0, 1, 1 ], "numerator": "2", "denominator": "2" }, { "coroot": [ 0, 0, 0, 1, 1, 0 ], "numerator": "2", "denominator": "2" }, { "coroot": [ 0, 0, 1, 1, 0, 0 ], "numerator": "2", "denominator": "2" }, { "coroot": [ 0, 1, 0, 1, 0, 0 ], "numerator": "2", "denominator": "2" }, { "coroot": [ 1, 0, 1, 0, 0, 0 ], "numerator": "3", "denominator": "2" }, { "coroot": [ 0, 0, 0, 1, 1, 1 ], "numerator": "3", "denominator": "3" }, { "coroot": [ 0, 0, 1, 1, 1, 0 ], "numerator": "3", "denominator": "3" }, { "coroot": [ 0, 1, 0, 1, 1, 0 ], "numerator": "3", "denominator": "3" }, { "coroot": [ 0, 1, 1, 1, 0, 0 ], "numerator": "3", "denominator": "3" }, { "coroot": [ 1, 0, 1, 1, 0, 0 ], "numerator": "4", "denominator": "3" }, { "coroot": [ 0, 0, 1, 1, 1, 1 ], "numerator": "4", "denominator": "4" }, { "coroot": [ 0, 1, 0, 1, 1, 1 ], "numerator": "4", "denominator": "4" }, { "coroot": [ 0, 1, 1, 1, 1, 0 ], "numerator": "4", "denominator": "4" }, { "coroot": [ 1, 0, 1, 1, 1, 0 ], "numerator": "5", "denominator": "4" }, { "coroot": [ 1, 1, 1, 1, 0, 0 ], "numerator": "5", "denominator": "4" }, { "coroot": [ 0, 1, 1, 1, 1, 1 ], "numerator": "5", "denominator": "5" }, { "coroot": [ 0, 1, 1, 2, 1, 0 ], "numerator": "5", "denominator": "5" }, { "coroot": [ 1, 0, 1, 1, 1, 1 ], "numerator": "6", "denominator": "5" }, { "coroot": [ 1, 1, 1, 1, 1, 0 ], "numerator": "6", "denominator": "5" }, { "coroot": [ 0, 1, 1, 2, 1, 1 ], "numerator": "6", "denominator": "6" }, { "coroot": [ 1, 1, 1, 1, 1, 1 ], "numerator": "7", "denominator": "6" }, { "coroot": [ 1, 1, 1, 2, 1, 0 ], "numerator": "7", "denominator": "6" }, { "coroot": [ 0, 1, 1, 2, 2, 1 ], "numerator": "7", "denominator": "7" }, { "coroot": [ 1, 1, 1, 2, 1, 1 ], "numerator": "8", "denominator": "7" }, { "coroot": [ 1, 1, 2, 2, 1, 0 ], "numerator": "8", "denominator": "7" }, { "coroot": [ 1, 1, 1, 2, 2, 1 ], "numerator": "9", "denominator": "8" }, { "coroot": [ 1, 1, 2, 2, 1, 1 ], "numerator": "9", "denominator": "8" }, { "coroot": [ 1, 1, 2, 2, 2, 1 ], "numerator": "10", "denominator": "9" }, { "coroot": [ 1, 1, 2, 3, 2, 1 ], "numerator": "11", "denominator": "10" }, { "coroot": [ 1, 2, 2, 3, 2, 1 ], "numerator": "12", "denominator": "11" } ], "algorithmVersion": "lie-representations-1.0.0", "category": "Finite-dimensional continuous complex irreducible representations of the compact simply connected group", "method": "Exact Weyl product over positive coroots; rational weight-metric Casimir; root-lattice descent; central-character Frobenius–Schur indicator.", "checks": { "positiveCorootCount": true, "dimensionIntegral": true, "weightGramSymmetric": true, "centralCharacterOrderDividesCenter": true }, "sources": [ { "id": "workbench-etingof-lie", "title": "Lie Groups and Lie Algebras I", "authors": [ "Pavel Etingof" ], "url": "https://math.mit.edu/~etingof/lnlg.pdf", "kind": "lecture-notes", "locatorHint": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-e", "title": "SageMath: Root system data for type E", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_E.html", "kind": "documentation", "locatorHint": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "E", "rank": "6", "labels": "1,0,0,0,0,0" } } } }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "slug": "e-6-compact-sc-weight-0-1-0-0-0-0", "name": "Adjoint of E6 (compact simply connected)", "symbol": "E6 · (0,1,0,0,0,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "E6 (compact simply connected)", "Adjoint", "78-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,1,0,0,0,0) of the specified simply connected compact group: complex dimension 78, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of E6 (compact simply connected) with Dynkin labels (0,1,0,0,0,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "78", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "12", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "12", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1,0,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. 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A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:e-6-compact-sc", "label": "Representation domain", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-e", "title": "SageMath: Root system data for type E", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_E.html", "kind": "documentation", "locatorHint": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "E", "rank": "6", "labels": "0,1,0,0,0,0" } } } }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "slug": "e-7-compact-sc-weight-0-0-0-0-0-0-1", "name": "56-dimensional representation of E7 (compact simply connected)", "symbol": "E7 · (0,0,0,0,0,0,1)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "E7 (compact simply connected)", "56-dimensional representation", "56-dimensional", "Quaternionic" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,0,0,0,0,0,1) of the specified simply connected compact group: complex dimension 56, quaternionic type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of E7 (compact simply connected) with Dynkin labels (0,0,0,0,0,0,1) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "56", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "57/4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "6", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Quaternionic", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "E", "rank": "7", "labels": "0,0,0,0,0,0,1" } } } }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "slug": "e-7-compact-sc-weight-1-0-0-0-0-0-0", "name": "Adjoint of E7 (compact simply connected)", "symbol": "E7 · (1,0,0,0,0,0,0)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "E7 (compact simply connected)", "Adjoint", "133-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (1,0,0,0,0,0,0) of the specified simply connected compact group: complex dimension 133, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of E7 (compact simply connected) with Dynkin labels (1,0,0,0,0,0,0) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "133", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "18", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "18", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0,0,0,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Use in a calculation", "paragraphs": [ { "text": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] 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1, 2, 2, 3, 2, 1, 0 ], "numerator": "12", "denominator": "11" }, { "coroot": [ 1, 1, 2, 3, 2, 2, 1 ], "numerator": "13", "denominator": "12" }, { "coroot": [ 1, 2, 2, 3, 2, 1, 1 ], "numerator": "13", "denominator": "12" }, { "coroot": [ 1, 1, 2, 3, 3, 2, 1 ], "numerator": "14", "denominator": "13" }, { "coroot": [ 1, 2, 2, 3, 2, 2, 1 ], "numerator": "14", "denominator": "13" }, { "coroot": [ 1, 2, 2, 3, 3, 2, 1 ], "numerator": "15", "denominator": "14" }, { "coroot": [ 1, 2, 2, 4, 3, 2, 1 ], "numerator": "16", "denominator": "15" }, { "coroot": [ 1, 2, 3, 4, 3, 2, 1 ], "numerator": "17", "denominator": "16" }, { "coroot": [ 2, 2, 3, 4, 3, 2, 1 ], "numerator": "19", "denominator": "17" } ], "algorithmVersion": "lie-representations-1.0.0", "category": "Finite-dimensional continuous complex irreducible representations of the compact simply connected group", "method": "Exact Weyl product over positive coroots; rational weight-metric Casimir; root-lattice descent; central-character Frobenius–Schur indicator.", "checks": { "positiveCorootCount": true, "dimensionIntegral": true, "weightGramSymmetric": true, "centralCharacterOrderDividesCenter": true }, "sources": [ { "id": "workbench-etingof-lie", "title": "Lie Groups and Lie Algebras I", "authors": [ "Pavel Etingof" ], "url": "https://math.mit.edu/~etingof/lnlg.pdf", "kind": "lecture-notes", "locatorHint": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-e", "title": "SageMath: Root system data for type E", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_E.html", "kind": "documentation", "locatorHint": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "E", "rank": "7", "labels": "1,0,0,0,0,0,0" } } } }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "slug": "e-8-compact-weight-0-0-0-0-0-0-0-1", "name": "Adjoint of E8 (compact simply connected)", "symbol": "E8 · (0,0,0,0,0,0,0,1)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "E8 (compact simply connected)", "Adjoint", "248-dimensional", "Real" ], "tags": [ "Highest weight", "Compact", "Irreducible" ], "summary": "Highest weight (0,0,0,0,0,0,0,1) of the specified simply connected compact group: complex dimension 248, real type, with exact dimension, Casimir, and center checks.", "definition": "The finite-dimensional continuous complex irreducible representation of E8 (compact simply connected) with Dynkin labels (0,0,0,0,0,0,0,1) in the displayed Bourbaki/Sage node order.", "definitionEvidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "properties": [ { "key": "dimension", "label": "Complex dimension", "value": "248", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "casimir", "label": "Quadratic Casimir C₂", "value": "30", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "index", "label": "Quadratic index T(R)", "value": "30", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,0,0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] }, { "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] }, { "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Highest weight and conventions", "paragraphs": [ { "text": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "global-form", "title": "Global form and center", "paragraphs": [ { "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] }, { "id": "physics", "title": "Use in a calculation", "paragraphs": [ { "text": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [] } ] } ], "relationships": [ { "type": "representation-of", "target": "group:e-8-compact", "label": "Representation domain", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional 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"coroot": [ 2, 3, 4, 5, 4, 3, 2, 1 ], "numerator": "25", "denominator": "24" }, { "coroot": [ 2, 3, 4, 6, 4, 3, 2, 1 ], "numerator": "26", "denominator": "25" }, { "coroot": [ 2, 3, 4, 6, 5, 3, 2, 1 ], "numerator": "27", "denominator": "26" }, { "coroot": [ 2, 3, 4, 6, 5, 4, 2, 1 ], "numerator": "28", "denominator": "27" }, { "coroot": [ 2, 3, 4, 6, 5, 4, 3, 1 ], "numerator": "29", "denominator": "28" }, { "coroot": [ 2, 3, 4, 6, 5, 4, 3, 2 ], "numerator": "31", "denominator": "29" } ], "algorithmVersion": "lie-representations-1.0.0", "category": "Finite-dimensional continuous complex irreducible representations of the compact simply connected group", "method": "Exact Weyl product over positive coroots; rational weight-metric Casimir; root-lattice descent; central-character Frobenius–Schur indicator.", "checks": { "positiveCorootCount": true, "dimensionIntegral": true, "weightGramSymmetric": true, "centralCharacterOrderDividesCenter": true }, "sources": [ { "id": "workbench-etingof-lie", "title": "Lie Groups and Lie Algebras I", "authors": [ "Pavel Etingof" ], "url": "https://math.mit.edu/~etingof/lnlg.pdf", "kind": "lecture-notes", "locatorHint": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-e", "title": "SageMath: Root system data for type E", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_E.html", "kind": "documentation", "locatorHint": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-weyl", "title": "SageMath: The Weyl Character Ring", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html", "kind": "documentation", "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-conrad-centers", "title": "Centers of Simply Connected Semisimple Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-adams-indicator", "title": "The Real Chevalley Involution", "authors": [ "Jeffrey Adams" ], "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf", "kind": "paper", "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." } ] }, "capabilityPresets": { "lie-representations": { "type": "E", "rank": "8", "labels": "0,0,0,0,0,0,0,1" } } } }, { "id": "algebra:standard-model-r", "slug": "standard-model-r", "name": "Compact Standard Model gauge algebra", "symbol": "su(3) ⊕ su(2) ⊕ u(1)", "category": "lie", "level": "reference", "aliases": [ "Standard Model algebra" ], "searchTerms": [ "Standard Model", "hypercharge", "reductive", "semisimple part" ], "tags": [ "Real algebra", "Compact", "Reductive" ], "summary": "The twelve-dimensional compact reductive real algebra su(3) ⊕ su(2) ⊕ u(1), shared by four specified matter-compatible compact global groups.", "definition": "The direct sum of the compact real algebras su(3), su(2), and u(1). Its real dimension is 8+3+1=12, its rank is 2+1+1=4, and its one-dimensional center makes it reductive but not semisimple.", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "center", "label": "Lie-algebra center", "value": "u(1), dimension one", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ], "sections": [ { "id": "global-forms", "title": "Global groups and charge normalization", "paragraphs": [ { "text": "The Lie algebra alone does not fix a connected global gauge group or the allowed line operators. The four group records here use the compact product SU(3) × SU(2) × U(1), integer charge Q=6Y, and subgroups of its specified diagonal Z6.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-factor", "target": "algebra:su-3-r", "label": "Color algebra", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-factor", "target": "algebra:su-2-r", "label": "Weak algebra", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-factor", "target": "algebra:u-1-r", "label": "Abelian algebra", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "integrates-to", "target": "group:standard-model-z1", "label": "Compact global form with quotient order 1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "integrates-to", "target": "group:standard-model-z2", "label": "Compact global form with quotient order 2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "integrates-to", "target": "group:standard-model-z3", "label": "Compact global form with quotient order 3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "integrates-to", "target": "group:standard-model-z6", "label": "Compact global form with quotient order 6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [], "kind": "algebra", "data": { "field": "R", "dimension": 12, "rank": 4, "simple": false, "displayTex": "\\mathfrak{su}(3)\\oplus\\mathfrak{su}(2)\\oplus\\mathfrak u(1)" } }, { "id": "group:standard-model-z1", "slug": "standard-model-z1", "name": "Standard Model compact covering product", "symbol": "SU(3) × SU(2) × U(1)", "category": "lie", "level": "executable", "aliases": [ "SM Z1", "Standard Model quotient 1" ], "searchTerms": [ "Standard Model", "hypercharge", "global form", "diagonal center", "SU5" ], "tags": [ "Compact", "Connected", "Product quotient" ], "summary": "A compact connected twelve-dimensional global group with algebra su(3) ⊕ su(2) ⊕ u(1), quotient order 1, and fundamental group Z. Usual Standard Model matter descends in the stated Q=6Y convention.", "definition": "The compact product SU(3) × SU(2) × U(1), with U(1) period 2π and integer hypercharge Q=6Y. It is a covering group of the three nontrivial specified quotients, but it is not the universal cover.", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group", "value": "Z", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "quotient-order", "label": "Specified quotient order", "value": "1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "center", "label": "Abstract center", "value": "U(1) × Z_6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "derivation": { "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.", "inputs": [ "source:conrad-pi1-lie", "Γd projects injectively to both the diagonal finite center and the U(1) factor." ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Matter and the embedded kernel", "paragraphs": [ { "text": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/1) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "text": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] }, { "id": "topology", "title": "Topology and the universal cover", "paragraphs": [ { "text": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_1 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_1. The compact product used as the quotient domain already has fundamental group Z.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Hypercharge and SU(5)", "paragraphs": [ { "text": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:standard-model-r", "label": "The common gauge algebra", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-central-quotient", "target": "group:standard-model-z2", "label": "Diagonal quotient of order 2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-central-quotient", "target": "group:standard-model-z3", "label": "Diagonal quotient of order 3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-central-quotient", "target": "group:standard-model-z6", "label": "Diagonal quotient of order 6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [ "product-gauge-groups", "local-operators", "operator-enumeration" ], "kind": "group", "data": { "category": "lie", "dimension": 12, "compact": true, "connected": true, "abelian": false, "simple": false, "algebraId": "algebra:standard-model-r", "displayTex": "SU(3)\\times SU(2)\\times U(1)", "quotientOrder": 1, "conventionId": "product-su-u1-center-v1", "quotient": { "continuous": false, "dimension": 0, "order": 1, "finiteOrder": 1, "invariantFactors": [], "latex": "1", "elementScope": "Complete finite subgroup.", "elements": [ { "centers": [ 0, 0 ], "phase": { "numerator": "0", "denominator": "1", "text": "0", "latex": "0" }, "order": 1 } ], "generators": [], "generatorMeaning": "One generating set; this is not an identification of each generator with a particular invariant factor.", "u1ProjectionOrder": 1 }, "fundamentalGroup": { "freeRank": 1, "torsionOrder": 1, "torsionInvariantFactors": [], "latex": "\\mathbb Z", "phaseZeroSubgroup": [ { "centers": [ 0, 0 ], "phase": { "numerator": "0", "denominator": "1", "text": "0", "latex": "0" }, "order": 1 } ], "derivation": "Lift the selected subgroup to the universal cover (product SU factors) × R. Its torsion is exactly the subgroup at U(1) phase zero; the real projection is an infinite cyclic lattice." }, "capabilityPresets": { "product-gauge-groups": { "title": "Standard Model matter with quotient order 1", "ranks": "3,2", "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3", "generators": "" }, "local-operators": { "setup": "{\"schema\":\"groups.qft.org/physical-setup/1\",\"name\":\"Weinberg operator with diagonal quotient order 1\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":1,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":3},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1}],\"operator\":{\"powers\":{\"L\":2,\"H\":2},\"derivatives\":0,\"equivalence\":\"algebraic\",\"coefficientField\":\"C\"}}" }, "operator-enumeration": { "inventory": "{\"schema\":\"groups.qft.org/operator-enumeration/1\",\"name\":\"Standard Model with diagonal quotient order 1\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":1,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"Q\",\"name\":\"Q\",\"kind\":\"left-weyl\",\"su3\":[1,0],\"su2\":1,\"charge\":1,\"flavors\":1},{\"id\":\"Qdag\",\"name\":\"Q†\",\"kind\":\"right-weyl\",\"su3\":[0,1],\"su2\":1,\"charge\":-1,\"flavors\":1},{\"id\":\"uc\",\"name\":\"uᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":-4,\"flavors\":1},{\"id\":\"ucdag\",\"name\":\"uᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":4,\"flavors\":1},{\"id\":\"dc\",\"name\":\"dᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":2,\"flavors\":1},{\"id\":\"dcdag\",\"name\":\"dᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":-2,\"flavors\":1},{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1},{\"id\":\"Ldag\",\"name\":\"L†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"ec\",\"name\":\"eᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":6,\"flavors\":1},{\"id\":\"ecdag\",\"name\":\"eᶜ†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":-6,\"flavors\":1},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"Hdag\",\"name\":\"H†\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1}],\"enumeration\":{\"minDimension\":1,\"maxDimension\":6,\"includeIdentity\":false,\"symmetries\":[]}}" } } } }, { "id": "group:standard-model-z2", "slug": "standard-model-z2", "name": "Standard Model global group with diagonal Z2 quotient", "symbol": "G_SM^(2)", "category": "lie", "level": "executable", "aliases": [ "SM Z2", "Standard Model quotient 2" ], "searchTerms": [ "Standard Model", "hypercharge", "global form", "diagonal center", "SU5" ], "tags": [ "Compact", "Connected", "Product quotient" ], "summary": "A compact connected twelve-dimensional global group with algebra su(3) ⊕ su(2) ⊕ u(1), quotient order 2, and fundamental group Z. Usual Standard Model matter descends in the stated Q=6Y convention.", "definition": "The group (SU(3) × SU(2) × U(1))/Γ_2, where Γ_2=⟨γ^3⟩ and γ=(exp(2πi/3) I₃, −I₂, exp(2πi/6)). The U(1) action on a matter multiplet is z^Q with integer Q=6Y.", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group", "value": "Z", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "quotient-order", "label": "Specified quotient order", "value": "2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "center", "label": "Abstract center", "value": "U(1) × Z_3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "derivation": { "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.", "inputs": [ "source:conrad-pi1-lie", "Γd projects injectively to both the diagonal finite center and the U(1) factor." ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Matter and the embedded kernel", "paragraphs": [ { "text": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/2) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "text": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] }, { "id": "topology", "title": "Topology and the universal cover", "paragraphs": [ { "text": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_2 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_2. The compact product used as the quotient domain already has fundamental group Z.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Hypercharge and SU(5)", "paragraphs": [ { "text": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:standard-model-r", "label": "The common gauge algebra", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "central-quotient-of", "target": "group:standard-model-z1", "label": "Specified compact covering product", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [ "product-gauge-groups", "local-operators", "operator-enumeration" ], "kind": "group", "data": { "category": "lie", "dimension": 12, "compact": true, "connected": true, "abelian": false, "simple": false, "algebraId": "algebra:standard-model-r", "displayTex": "G_{\\mathrm{SM}}^{(2)}", "quotientOrder": 2, "conventionId": "product-su-u1-center-v1", "quotient": { "continuous": false, "dimension": 0, "order": 2, "finiteOrder": 2, "invariantFactors": [ 2 ], "latex": "\\mathbb Z_{2}", "elementScope": "Complete finite subgroup.", "elements": [ { "centers": [ 0, 0 ], "phase": { "numerator": "0", "denominator": "1", "text": "0", "latex": "0" }, "order": 1 }, { "centers": [ 0, 1 ], "phase": { "numerator": "1", "denominator": "2", "text": "1/2", "latex": "\\frac{1}{2}" }, "order": 2 } ], "generators": [ { "centers": [ 0, 1 ], "phase": { "numerator": "1", "denominator": "2", "text": "1/2", "latex": "\\frac{1}{2}" }, "order": 2 } ], "generatorMeaning": "One generating set; this is not an identification of each generator with a particular invariant factor.", "u1ProjectionOrder": 2 }, "fundamentalGroup": { "freeRank": 1, "torsionOrder": 1, "torsionInvariantFactors": [], "latex": "\\mathbb Z", "phaseZeroSubgroup": [ { "centers": [ 0, 0 ], "phase": { "numerator": "0", "denominator": "1", "text": "0", "latex": "0" }, "order": 1 } ], "derivation": "Lift the selected subgroup to the universal cover (product SU factors) × R. Its torsion is exactly the subgroup at U(1) phase zero; the real projection is an infinite cyclic lattice." }, "capabilityPresets": { "product-gauge-groups": { "title": "Standard Model matter with quotient order 2", "ranks": "3,2", "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3", "generators": "0,1 | 1/2" }, "local-operators": { "setup": "{\"schema\":\"groups.qft.org/physical-setup/1\",\"name\":\"Weinberg operator with diagonal quotient order 2\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":2,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":3},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1}],\"operator\":{\"powers\":{\"L\":2,\"H\":2},\"derivatives\":0,\"equivalence\":\"algebraic\",\"coefficientField\":\"C\"}}" }, "operator-enumeration": { "inventory": "{\"schema\":\"groups.qft.org/operator-enumeration/1\",\"name\":\"Standard Model with diagonal quotient order 2\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":2,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"Q\",\"name\":\"Q\",\"kind\":\"left-weyl\",\"su3\":[1,0],\"su2\":1,\"charge\":1,\"flavors\":1},{\"id\":\"Qdag\",\"name\":\"Q†\",\"kind\":\"right-weyl\",\"su3\":[0,1],\"su2\":1,\"charge\":-1,\"flavors\":1},{\"id\":\"uc\",\"name\":\"uᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":-4,\"flavors\":1},{\"id\":\"ucdag\",\"name\":\"uᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":4,\"flavors\":1},{\"id\":\"dc\",\"name\":\"dᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":2,\"flavors\":1},{\"id\":\"dcdag\",\"name\":\"dᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":-2,\"flavors\":1},{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1},{\"id\":\"Ldag\",\"name\":\"L†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"ec\",\"name\":\"eᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":6,\"flavors\":1},{\"id\":\"ecdag\",\"name\":\"eᶜ†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":-6,\"flavors\":1},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"Hdag\",\"name\":\"H†\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1}],\"enumeration\":{\"minDimension\":1,\"maxDimension\":6,\"includeIdentity\":false,\"symmetries\":[]}}" } } } }, { "id": "group:standard-model-z3", "slug": "standard-model-z3", "name": "Standard Model global group with diagonal Z3 quotient", "symbol": "G_SM^(3)", "category": "lie", "level": "executable", "aliases": [ "SM Z3", "Standard Model quotient 3" ], "searchTerms": [ "Standard Model", "hypercharge", "global form", "diagonal center", "SU5" ], "tags": [ "Compact", "Connected", "Product quotient" ], "summary": "A compact connected twelve-dimensional global group with algebra su(3) ⊕ su(2) ⊕ u(1), quotient order 3, and fundamental group Z. Usual Standard Model matter descends in the stated Q=6Y convention.", "definition": "The group (SU(3) × SU(2) × U(1))/Γ_3, where Γ_3=⟨γ^2⟩ and γ=(exp(2πi/3) I₃, −I₂, exp(2πi/6)). The U(1) action on a matter multiplet is z^Q with integer Q=6Y.", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group", "value": "Z", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "quotient-order", "label": "Specified quotient order", "value": "3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "center", "label": "Abstract center", "value": "U(1) × Z_2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "derivation": { "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.", "inputs": [ "source:conrad-pi1-lie", "Γd projects injectively to both the diagonal finite center and the U(1) factor." ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Matter and the embedded kernel", "paragraphs": [ { "text": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/3) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "text": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] }, { "id": "topology", "title": "Topology and the universal cover", "paragraphs": [ { "text": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_3 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_3. The compact product used as the quotient domain already has fundamental group Z.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Hypercharge and SU(5)", "paragraphs": [ { "text": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:standard-model-r", "label": "The common gauge algebra", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "central-quotient-of", "target": "group:standard-model-z1", "label": "Specified compact covering product", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [ "product-gauge-groups", "local-operators", "operator-enumeration" ], "kind": "group", "data": { "category": "lie", "dimension": 12, "compact": true, "connected": true, "abelian": false, "simple": false, "algebraId": "algebra:standard-model-r", "displayTex": "G_{\\mathrm{SM}}^{(3)}", "quotientOrder": 3, "conventionId": "product-su-u1-center-v1", "quotient": { "continuous": false, "dimension": 0, "order": 3, "finiteOrder": 3, "invariantFactors": [ 3 ], "latex": "\\mathbb Z_{3}", "elementScope": "Complete finite subgroup.", "elements": [ { "centers": [ 0, 0 ], "phase": { "numerator": "0", "denominator": "1", "text": "0", "latex": "0" }, "order": 1 }, { "centers": [ 2, 0 ], "phase": { "numerator": "1", "denominator": "3", "text": "1/3", "latex": "\\frac{1}{3}" }, "order": 3 }, { "centers": [ 1, 0 ], "phase": { "numerator": "2", "denominator": "3", "text": "2/3", "latex": "\\frac{2}{3}" }, "order": 3 } ], "generators": [ { "centers": [ 2, 0 ], "phase": { "numerator": "1", "denominator": "3", "text": "1/3", "latex": "\\frac{1}{3}" }, "order": 3 } ], "generatorMeaning": "One generating set; this is not an identification of each generator with a particular invariant factor.", "u1ProjectionOrder": 3 }, "fundamentalGroup": { "freeRank": 1, "torsionOrder": 1, "torsionInvariantFactors": [], "latex": "\\mathbb Z", "phaseZeroSubgroup": [ { "centers": [ 0, 0 ], "phase": { "numerator": "0", "denominator": "1", "text": "0", "latex": "0" }, "order": 1 } ], "derivation": "Lift the selected subgroup to the universal cover (product SU factors) × R. Its torsion is exactly the subgroup at U(1) phase zero; the real projection is an infinite cyclic lattice." }, "capabilityPresets": { "product-gauge-groups": { "title": "Standard Model matter with quotient order 3", "ranks": "3,2", "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3", "generators": "2,0 | 1/3" }, "local-operators": { "setup": "{\"schema\":\"groups.qft.org/physical-setup/1\",\"name\":\"Weinberg operator with diagonal quotient order 3\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":3,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":3},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1}],\"operator\":{\"powers\":{\"L\":2,\"H\":2},\"derivatives\":0,\"equivalence\":\"algebraic\",\"coefficientField\":\"C\"}}" }, "operator-enumeration": { "inventory": "{\"schema\":\"groups.qft.org/operator-enumeration/1\",\"name\":\"Standard Model with diagonal quotient order 3\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":3,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"Q\",\"name\":\"Q\",\"kind\":\"left-weyl\",\"su3\":[1,0],\"su2\":1,\"charge\":1,\"flavors\":1},{\"id\":\"Qdag\",\"name\":\"Q†\",\"kind\":\"right-weyl\",\"su3\":[0,1],\"su2\":1,\"charge\":-1,\"flavors\":1},{\"id\":\"uc\",\"name\":\"uᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":-4,\"flavors\":1},{\"id\":\"ucdag\",\"name\":\"uᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":4,\"flavors\":1},{\"id\":\"dc\",\"name\":\"dᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":2,\"flavors\":1},{\"id\":\"dcdag\",\"name\":\"dᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":-2,\"flavors\":1},{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1},{\"id\":\"Ldag\",\"name\":\"L†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"ec\",\"name\":\"eᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":6,\"flavors\":1},{\"id\":\"ecdag\",\"name\":\"eᶜ†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":-6,\"flavors\":1},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"Hdag\",\"name\":\"H†\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1}],\"enumeration\":{\"minDimension\":1,\"maxDimension\":6,\"includeIdentity\":false,\"symmetries\":[]}}" } } } }, { "id": "group:standard-model-z6", "slug": "standard-model-z6", "name": "Standard Model global group with diagonal Z6 quotient", "symbol": "G_SM^(6)", "category": "lie", "level": "executable", "aliases": [ "SM Z6", "Standard Model quotient 6", "S(U(3) × U(2))" ], "searchTerms": [ "Standard Model", "hypercharge", "global form", "diagonal center", "SU5" ], "tags": [ "Compact", "Connected", "Product quotient" ], "summary": "A compact connected twelve-dimensional global group with algebra su(3) ⊕ su(2) ⊕ u(1), quotient order 6, and fundamental group Z. Usual Standard Model matter descends in the stated Q=6Y convention.", "definition": "The group (SU(3) × SU(2) × U(1))/Γ_6, where Γ_6=⟨γ^1⟩ and γ=(exp(2πi/3) I₃, −I₂, exp(2πi/6)). The U(1) action on a matter multiplet is z^Q with integer Q=6Y.", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "pi1", "label": "Fundamental group", "value": "Z", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "quotient-order", "label": "Specified quotient order", "value": "6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "center", "label": "Abstract center", "value": "U(1)", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "derivation": { "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.", "inputs": [ "source:conrad-pi1-lie", "Γd projects injectively to both the diagonal finite center and the U(1) factor." ] }, "assumptions": [] } ], "sections": [ { "id": "representations", "title": "Matter and the embedded kernel", "paragraphs": [ { "text": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/6) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "text": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] }, { "id": "topology", "title": "Topology and the universal cover", "paragraphs": [ { "text": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_6 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_6. The compact product used as the quotient domain already has fundamental group Z.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] }, { "id": "conventions", "title": "Hypercharge and SU(5)", "paragraphs": [ { "text": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-algebra", "target": "algebra:standard-model-r", "label": "The common gauge algebra", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "central-quotient-of", "target": "group:standard-model-z1", "label": "Specified compact covering product", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "embeds-in", "target": "group:su-5", "label": "Block subgroup of SU(5) with the specified map", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [ "product-gauge-groups", "local-operators", "operator-enumeration" ], "kind": "group", "data": { "category": "lie", "dimension": 12, "compact": true, "connected": true, "abelian": false, "simple": false, "algebraId": "algebra:standard-model-r", "displayTex": "G_{\\mathrm{SM}}^{(6)}", "quotientOrder": 6, "conventionId": "product-su-u1-center-v1", "quotient": { "continuous": false, "dimension": 0, "order": 6, "finiteOrder": 6, "invariantFactors": [ 6 ], "latex": "\\mathbb Z_{6}", "elementScope": "Complete finite subgroup.", "elements": [ { "centers": [ 0, 0 ], "phase": { "numerator": "0", "denominator": "1", "text": "0", "latex": "0" }, "order": 1 }, { "centers": [ 1, 1 ], "phase": { "numerator": "1", "denominator": "6", "text": "1/6", "latex": "\\frac{1}{6}" }, "order": 6 }, { "centers": [ 2, 0 ], "phase": { "numerator": "1", "denominator": "3", "text": "1/3", "latex": "\\frac{1}{3}" }, "order": 3 }, { "centers": [ 0, 1 ], "phase": { "numerator": "1", "denominator": "2", "text": "1/2", "latex": "\\frac{1}{2}" }, "order": 2 }, { "centers": [ 1, 0 ], "phase": { "numerator": "2", "denominator": "3", "text": "2/3", "latex": "\\frac{2}{3}" }, "order": 3 }, { "centers": [ 2, 1 ], "phase": { "numerator": "5", "denominator": "6", "text": "5/6", "latex": "\\frac{5}{6}" }, "order": 6 } ], "generators": [ { "centers": [ 1, 1 ], "phase": { "numerator": "1", "denominator": "6", "text": "1/6", "latex": "\\frac{1}{6}" }, "order": 6 } ], "generatorMeaning": "One generating set; this is not an identification of each generator with a particular invariant factor.", "u1ProjectionOrder": 6 }, "fundamentalGroup": { "freeRank": 1, "torsionOrder": 1, "torsionInvariantFactors": [], "latex": "\\mathbb Z", "phaseZeroSubgroup": [ { "centers": [ 0, 0 ], "phase": { "numerator": "0", "denominator": "1", "text": "0", "latex": "0" }, "order": 1 } ], "derivation": "Lift the selected subgroup to the universal cover (product SU factors) × R. Its torsion is exactly the subgroup at U(1) phase zero; the real projection is an infinite cyclic lattice." }, "capabilityPresets": { "product-gauge-groups": { "title": "Standard Model matter with quotient order 6", "ranks": "3,2", "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3", "generators": "1,1 | 1/6" }, "local-operators": { "setup": "{\"schema\":\"groups.qft.org/physical-setup/1\",\"name\":\"Weinberg operator with diagonal quotient order 6\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":6,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":3},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1}],\"operator\":{\"powers\":{\"L\":2,\"H\":2},\"derivatives\":0,\"equivalence\":\"algebraic\",\"coefficientField\":\"C\"}}" }, "operator-enumeration": { "inventory": "{\"schema\":\"groups.qft.org/operator-enumeration/1\",\"name\":\"Standard Model with diagonal quotient order 6\",\"spacetime\":{\"dimension\":4,\"signature\":\"Lorentzian\"},\"gauge\":{\"kind\":\"su3-su2-u1\",\"quotientOrder\":6,\"u1Normalization\":\"integer-charge-period-2pi\"},\"fields\":[{\"id\":\"Q\",\"name\":\"Q\",\"kind\":\"left-weyl\",\"su3\":[1,0],\"su2\":1,\"charge\":1,\"flavors\":1},{\"id\":\"Qdag\",\"name\":\"Q†\",\"kind\":\"right-weyl\",\"su3\":[0,1],\"su2\":1,\"charge\":-1,\"flavors\":1},{\"id\":\"uc\",\"name\":\"uᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":-4,\"flavors\":1},{\"id\":\"ucdag\",\"name\":\"uᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":4,\"flavors\":1},{\"id\":\"dc\",\"name\":\"dᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,1],\"su2\":0,\"charge\":2,\"flavors\":1},{\"id\":\"dcdag\",\"name\":\"dᶜ†\",\"kind\":\"right-weyl\",\"su3\":[1,0],\"su2\":0,\"charge\":-2,\"flavors\":1},{\"id\":\"L\",\"name\":\"L\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1},{\"id\":\"Ldag\",\"name\":\"L†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"ec\",\"name\":\"eᶜ\",\"kind\":\"left-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":6,\"flavors\":1},{\"id\":\"ecdag\",\"name\":\"eᶜ†\",\"kind\":\"right-weyl\",\"su3\":[0,0],\"su2\":0,\"charge\":-6,\"flavors\":1},{\"id\":\"H\",\"name\":\"H\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":3,\"flavors\":1},{\"id\":\"Hdag\",\"name\":\"H†\",\"kind\":\"scalar\",\"su3\":[0,0],\"su2\":1,\"charge\":-3,\"flavors\":1}],\"enumeration\":{\"minDimension\":1,\"maxDimension\":6,\"includeIdentity\":false,\"symmetries\":[]}}" } } } }, { "id": "map:standard-model-cover-to-z2", "slug": "standard-model-cover-to-z2", "name": "Standard Model quotient map of order 2", "symbol": "G_SM^(1) → G_SM^(2)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Standard Model", "quotient", "cover", "hypercharge" ], "tags": [ "Central quotient", "Surjective" ], "summary": "The explicit quotient by the subgroup generated by γ^3, with γ=(ω₃,−1,exp(2πi/6)) and integer Q=6Y.", "definition": "The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_2=⟨γ^3⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 2.", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "kernel", "label": "Kernel", "value": "Γ_2 = ⟨γ^3⟩ ≅ Z_2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ], "sections": [ { "id": "matter", "title": "Matter descent", "paragraphs": [ { "text": "A representation of the compact product descends exactly when every generator of this embedded kernel acts trivially. Test the color center, weak center, and abelian phase together; the same abstract cyclic group embedded in one factor can give a different answer.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:standard-model-z1", "label": "Compact covering product", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-codomain", "target": "group:standard-model-z2", "label": "Chosen global form", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [ "product-gauge-groups" ], "kind": "map", "data": { "domain": "group:standard-model-z1", "codomain": "group:standard-model-z2", "kernel": "Γ_2 = ⟨γ^3⟩", "image": "G_SM^(2)", "injective": false, "surjective": true, "displayTex": "G_{\\mathrm{SM}}^{(1)}\\longrightarrow G_{\\mathrm{SM}}^{(2)}", "capabilityPresets": { "product-gauge-groups": { "title": "Standard Model matter with quotient order 2", "ranks": "3,2", "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3", "generators": "0,1 | 1/2" } } } }, { "id": "map:standard-model-cover-to-z3", "slug": "standard-model-cover-to-z3", "name": "Standard Model quotient map of order 3", "symbol": "G_SM^(1) → G_SM^(3)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Standard Model", "quotient", "cover", "hypercharge" ], "tags": [ "Central quotient", "Surjective" ], "summary": "The explicit quotient by the subgroup generated by γ^2, with γ=(ω₃,−1,exp(2πi/6)) and integer Q=6Y.", "definition": "The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_3=⟨γ^2⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 3.", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "kernel", "label": "Kernel", "value": "Γ_3 = ⟨γ^2⟩ ≅ Z_3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ], "sections": [ { "id": "matter", "title": "Matter descent", "paragraphs": [ { "text": "A representation of the compact product descends exactly when every generator of this embedded kernel acts trivially. Test the color center, weak center, and abelian phase together; the same abstract cyclic group embedded in one factor can give a different answer.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:standard-model-z1", "label": "Compact covering product", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-codomain", "target": "group:standard-model-z3", "label": "Chosen global form", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [ "product-gauge-groups" ], "kind": "map", "data": { "domain": "group:standard-model-z1", "codomain": "group:standard-model-z3", "kernel": "Γ_3 = ⟨γ^2⟩", "image": "G_SM^(3)", "injective": false, "surjective": true, "displayTex": "G_{\\mathrm{SM}}^{(1)}\\longrightarrow G_{\\mathrm{SM}}^{(3)}", "capabilityPresets": { "product-gauge-groups": { "title": "Standard Model matter with quotient order 3", "ranks": "3,2", "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3", "generators": "2,0 | 1/3" } } } }, { "id": "map:standard-model-cover-to-z6", "slug": "standard-model-cover-to-z6", "name": "Standard Model quotient map of order 6", "symbol": "G_SM^(1) → G_SM^(6)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "Standard Model", "quotient", "cover", "hypercharge" ], "tags": [ "Central quotient", "Surjective" ], "summary": "The explicit quotient by the subgroup generated by γ^1, with γ=(ω₃,−1,exp(2πi/6)) and integer Q=6Y.", "definition": "The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_6=⟨γ^1⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 6.", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "kernel", "label": "Kernel", "value": "Γ_6 = ⟨γ^1⟩ ≅ Z_6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ], "sections": [ { "id": "matter", "title": "Matter descent", "paragraphs": [ { "text": "A representation of the compact product descends exactly when every generator of this embedded kernel acts trivially. Test the color center, weak center, and abelian phase together; the same abstract cyclic group embedded in one factor can give a different answer.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:standard-model-z1", "label": "Compact covering product", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-codomain", "target": "group:standard-model-z6", "label": "Chosen global form", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [ "product-gauge-groups" ], "kind": "map", "data": { "domain": "group:standard-model-z1", "codomain": "group:standard-model-z6", "kernel": "Γ_6 = ⟨γ^1⟩", "image": "G_SM^(6)", "injective": false, "surjective": true, "displayTex": "G_{\\mathrm{SM}}^{(1)}\\longrightarrow G_{\\mathrm{SM}}^{(6)}", "capabilityPresets": { "product-gauge-groups": { "title": "Standard Model matter with quotient order 6", "ranks": "3,2", "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3", "generators": "1,1 | 1/6" } } } }, { "id": "map:su-3-su-2-u1-to-su-5-block", "slug": "su-3-su-2-u1-to-su-5-block", "name": "Color-first SU(5) block homomorphism", "symbol": "SU(3) × SU(2) × U(1)_X → SU(5)", "category": "lie", "level": "executable", "aliases": [], "searchTerms": [ "SU5", "hypercharge", "branching", "block subgroup" ], "tags": [ "Homomorphism", "Finite kernel", "Charge refined" ], "summary": "The block map diag(z² A,z⁻³ B), with primitive charge X and kernel Z6. Physical hypercharge is Y=−X/6.", "definition": "The homomorphism ι(A,B,z)=diag(z² A,z⁻³ B) from SU(3) × SU(2) × U(1) to SU(5), with the color block first. Its kernel is generated by (exp(−2πi/3) I₃,−I₂,exp(2πi/6)) and its image is S(U(3) × U(2)).", "definitionEvidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "properties": [ { "key": "kernel", "label": "Kernel", "value": "Z6 with the explicitly stated generator", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "image", "label": "Image", "value": "S(U(3) × U(2))", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. 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Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [ "charged-branching" ], "kind": "map", "data": { "domain": "group:standard-model-z1", "codomain": "group:su-5", "kernel": "⟨(ω₃⁻¹,−1,exp(2πi/6))⟩ ≅ Z6", "image": "S(U(3) × U(2))", "injective": false, "surjective": false, "displayTex": "SU(3)\\times SU(2)\\times U(1)_X\\longrightarrow SU(5)", "conventionId": "su-n-block-u1-v1", "capabilityPresets": { "charged-branching": { "n": "5", "p": "3", "labels": "0,1,0,0" } } } }, { "id": "map:standard-model-z6-into-su5", "slug": "standard-model-z6-into-su5", "name": "The diagonal-Z6 Standard Model group inside SU(5)", "symbol": "G_SM^(6) ↪ SU(5)", "category": "lie", "level": "reference", "aliases": [], "searchTerms": [ "Standard Model", "SU5", "embedding", "hypercharge" ], "tags": [ "Embedding", "Block subgroup" ], "summary": "The injective group map induced by diag(z⁻² A,z³ B) in the Q=6Y convention.", "definition": "The map [(A,B,z)] ↦ diag(z⁻² A,z³ B) from G_SM^(6) to SU(5). 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Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] }, { "key": "injective", "label": "Injective", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. 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Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [] } ] } ], "relationships": [ { "type": "has-domain", "target": "group:standard-model-z6", "label": "The specified diagonal quotient", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "has-codomain", "target": "group:su-5", "label": "Parent compact group", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] }, { "type": "related-map", "target": "map:su-3-su-2-u1-to-su-5-block", "label": "Computable block map in the X convention", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ] } ], "capabilities": [], "kind": "map", "data": { "domain": "group:standard-model-z6", "codomain": "group:su-5", "kernel": "1", "image": "S(U(3) × U(2))", "injective": true, "surjective": false, "displayTex": "G_{\\mathrm{SM}}^{(6)}\\hookrightarrow SU(5)", "conventionId": "product-su-u1-center-v1" } } ], "sources": [ { "id": "zuber", "title": "Zuber · Invariances in Physics and Group Theory", "authors": [], "url": "https://www.phys.ens.psl.eu/sites/default/files/2024-09/zuber_invariances_and_group_theory.pdf", "kind": "lecture-notes", "licenseNote": "Linked source with original summaries of mathematical facts. 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Craven" ], "url": "https://doi.org/10.1515/jgth-2018-0014", "kind": "paper", "locatorHint": "Section 7, seven exceptional isomorphisms before Proposition 24", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-benvenuti-delta27", "title": "Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics", "authors": [ "Sergio Benvenuti", "Bo Feng", "Amihay Hanany", "Yang-Hui He" ], "url": "https://arxiv.org/abs/hep-th/0608050", "kind": "paper", "locatorHint": "Section 3, equation (3.12): Delta(27) matrix generators and Heisenberg identification", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-sporadic-index", "title": "ATLAS of Finite Group Representations: Sporadic groups", "authors": [ "Robert A. Wilson", "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/", "kind": "database", "locatorHint": "Sporadic group index; Tits group listed separately under Miscellaneous", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-c-2", "title": "GroupNames: C₂", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C2.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-c-4", "title": "GroupNames: C₄", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C4.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-c-5", "title": "GroupNames: C₅", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C5.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-c-6", "title": "GroupNames: C₆", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C6.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-v-4", "title": "GroupNames: C₂ × C₂", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C2%5E2.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-d-10", "title": "GroupNames: D₅, order 10", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/D5.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-d-12", "title": "GroupNames: D₆, order 12", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/D6.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-dic-3", "title": "GroupNames: Dic₃", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/Dic3.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-q-16", "title": "GroupNames: Q₁₆", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/Q16.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-binary-tetrahedral", "title": "GroupNames: SL(2,3), binary tetrahedral", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/SL(2,3).html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-binary-octahedral", "title": "GroupNames: CSU(2,3), binary octahedral", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/CSU(2,3).html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-binary-icosahedral", "title": "GroupNames: SL(2,5), binary icosahedral", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/97/SL(2,5).html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-psl-2-7", "title": "GroupNames: GL(3,2) ≅ PSL(2,7)", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/163/GL(3,2).html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-delta-27", "title": "GroupNames: He₃", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/He3.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-groupnames-a-6", "title": "GroupNames: A₆", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/356/A6.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-m11", "title": "ATLAS: Mathieu group M11", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-m12", "title": "ATLAS: Mathieu group M12", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-m22", "title": "ATLAS: Mathieu group M22", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M22/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-m23", "title": "ATLAS: Mathieu group M23", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M23/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-m24", "title": "ATLAS: Mathieu group M24", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M24/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-j1", "title": "ATLAS: First Janko group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J1/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-j2", "title": "ATLAS: Hall–Janko group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J2/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-j3", "title": "ATLAS: Third Janko group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J3/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-j4", "title": "ATLAS: Fourth Janko group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J4/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-co1", "title": "ATLAS: First Conway group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-co2", "title": "ATLAS: Second Conway group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co2/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-co3", "title": "ATLAS: Third Conway group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co3/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-fi22", "title": "ATLAS: Fischer group Fi22", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/F22/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-fi23", "title": "ATLAS: Fischer group Fi23", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/F23/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-fi24-prime", "title": "ATLAS: Fischer simple group Fi24 prime", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/F24/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-hs", "title": "ATLAS: Higman–Sims group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/HS/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-mcl", "title": "ATLAS: McLaughlin group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/McL/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-he", "title": "ATLAS: Held group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-ru", "title": "ATLAS: Rudvalis group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Ru/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-suz", "title": "ATLAS: Sporadic Suzuki group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Suz/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-on", "title": "ATLAS: O’Nan group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ON/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-hn", "title": "ATLAS: Harada–Norton group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/HN/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-ly", "title": "ATLAS: Lyons group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Ly/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-th", "title": "ATLAS: Thompson group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Th/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-b", "title": "ATLAS: Baby Monster group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/B/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-m", "title": "ATLAS: Monster group", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M/", "kind": "database", "locatorHint": "Heading; Order; Standard generators; Representations", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "finite-atlas-tits", "title": "ATLAS: Tits group ²F₄(2)′", "authors": [ "ATLAS of Finite Group Representations contributors" ], "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TF42/", "kind": "database", "locatorHint": "Order; standard generators; ²F₄(2) extension", "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables." }, { "id": "conrad-centers", "title": "Math 210C. Centers of simply connected semisimple groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf", "kind": "lecture-notes", "locatorHint": "Center tables on pages 2–4; low-rank isomorphisms on page 4.", "licenseNote": "Linked author-hosted handout; factual center data are restated in original notation." }, { "id": "etingof-lie-notes", "title": "Lie Groups and Lie Algebras", "authors": [ "Pavel Etingof" ], "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec_full.pdf", "kind": "lecture-notes", "locatorHint": "MIT 18.755, Spring 2024. Cite section numbers and printed pages.", "licenseNote": "Linked MIT OpenCourseWare lecture notes. Original summaries only; website content is not offered as a reproduction of these notes." }, { "id": "etingof-coxeter", "title": "Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual Representations", "authors": [ "Pavel Etingof" ], "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec06.pdf", "kind": "lecture-notes", "locatorHint": "Sections 32.1–32.5, printed pages 172–178.", "licenseNote": "Linked MIT OpenCourseWare notes. Numerical root-system facts are used with original explanations." }, { "id": "etingof-real-forms", "title": "Lecture 15: Classification of Real Forms of Semisimple Lie Algebras", "authors": [ "Pavel Etingof" ], "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec15.pdf", "kind": "lecture-notes", "locatorHint": "Section 41, printed pages 218–225.", "licenseNote": "Linked MIT OpenCourseWare notes; definitions and classification facts are summarized in original prose." }, { "id": "etingof-global-forms", "title": "Lecture 17: Classification of Connected Compact and Complex Reductive Groups", "authors": [ "Pavel Etingof" ], "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec17.pdf", "kind": "lecture-notes", "locatorHint": "Sections 43.1–43.2, printed pages 230–235.", "licenseNote": "Linked MIT OpenCourseWare notes; global-form and representation criteria are restated." }, { "id": "knapp-structure", "title": "Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple Groups", "authors": [ "Anthony W. 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Only introductory structural facts are summarized." }, { "id": "hatcher-algebraic-topology", "title": "Algebraic Topology", "authors": [ "Allen Hatcher" ], "url": "https://pi.math.cornell.edu/~hatcher/AT/AT.pdf", "kind": "book", "locatorHint": "Section 4.2, Theorem 4.41 (printed page 376) and the preceding Hopf-fibration example.", "licenseNote": "Linked author-hosted textbook. Exact sequences below are independently derived from the stated theorem." }, { "id": "finite-etingof", "title": "Etingof et al., Introduction to Representation Theory", "authors": [], "url": "https://math.mit.edu/~etingof/reprbook.pdf", "kind": "lecture-notes", "locatorHint": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators", "licenseNote": "Linked authoritative source for checking mathematical facts. Original table specification and explanations are maintained here; this release does not redistribute an upstream database or assert a license over its content." }, { "id": "finite-gap-power-maps", "title": "GAP Reference Manual, Chapter 73: Maps Concerning Character Tables", "authors": [], "url": "https://gap-system.github.io/gap/doc/ref/chap73_mj.html", "kind": "documentation", "locatorHint": "Section 73.1: power maps are extra class data and need not follow from the character matrix alone", "licenseNote": "Linked authoritative source for checking mathematical facts. Original table specification and explanations are maintained here; this release does not redistribute an upstream database or assert a license over its content." }, { "id": "workbench-etingof-lie", "title": "Lie Groups and Lie Algebras I", "authors": [ "Pavel Etingof" ], "url": "https://math.mit.edu/~etingof/lnlg.pdf", "kind": "lecture-notes", "locatorHint": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-roots", "title": "SageMath: Cartan matrices", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html", "kind": "documentation", "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.", "licenseNote": "Linked primary source. 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No upstream implementation or document text is redistributed." }, { "id": "workbench-lieart", "title": "LieART — A Mathematica Application for Lie Algebras and Representation Theory", "authors": [ "Robert Feger", "Thomas W. Kephart" ], "url": "https://arxiv.org/abs/1206.6379", "kind": "paper", "locatorHint": "Section 4.5.3 and Tables A.37–A.39: Young-tableau tensor methods and SU(4), SU(5), SU(6) fixtures.", "licenseNote": "Linked primary mathematical or software documentation supporting original algorithms and prose. No upstream implementation or document text is redistributed." }, { "id": "block-baez-huerta", "title": "The Algebra of Grand Unified Theories", "authors": [ "John C. Baez", "John Huerta" ], "url": "https://arxiv.org/abs/0904.1556", "kind": "paper", "locatorHint": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.", "licenseNote": "Linked primary source supporting original derivations and calculations. No upstream text, software, or representation dataset is redistributed." }, { "id": "conrad-pi1-lie", "title": "Brian Conrad, Fundamental Groups of Lie Groups", "authors": [ "Brian Conrad" ], "url": "https://math.stanford.edu/~conrad/210CPage/handouts/pi1stuff.pdf", "kind": "lecture-notes", "locatorHint": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup", "licenseNote": "Linked primary source; the product-group specialization is derived in the local method page." }, { "id": "marinissen-rahn-waalewijn-2020", "title": "Efficient Hilbert Series for Effective Theories", "authors": [ "Coenraad B. Marinissen", "Rudi Rahn", "Wouter J. Waalewijn" ], "url": "https://arxiv.org/abs/2004.09521", "kind": "paper", "locatorHint": "§II A, Eqs. (2)–(4): symmetric/exterior powers and Haar projection without derivatives.", "licenseNote": "Linked primary research paper; the implementation and explanatory text are original. No upstream code or figures are redistributed." }, { "id": "lehman-martin-2015", "title": "Hilbert Series for Constructing Lagrangians: expanding the phenomenologist’s toolbox", "authors": [ "Landon Lehman", "Adam Martin" ], "url": "https://arxiv.org/abs/1503.07537", "kind": "paper", "locatorHint": "§4.1, Eq. (4.2): 57 QQQL operators for three generations; Appendix A: characters, plethystic exponentials and Haar measure.", "licenseNote": "Linked primary research paper; benchmark values are independently recomputed." }, { "id": "fonseca-2020", "title": "Enumerating the operators of an effective field theory", "authors": [ "Renato M. Fonseca" ], "url": "https://arxiv.org/abs/1907.12584", "kind": "paper", "locatorHint": "§3, Eq. (49) and Table 2: QQQL flavor count n_L n_Q (2 n_Q² + 1)/3; Eq. (51): n²(n²−1)/12 four-sterile-Weyl invariants.", "licenseNote": "Linked primary research paper; original independent character calculations reproduce its flavor polynomial." }, { "id": "dreiner-haber-martin-2010", "title": "Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry", "authors": [ "Herbi K. Dreiner", "Howard E. Haber", "Stephen P. 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The local enumeration and explanatory derivation are original; no upstream text, code, or figures are redistributed." }, { "id": "alonso-et-al-2014-baryon-violation", "title": "Renormalization group evolution of dimension-six baryon number violating operators", "authors": [ "Rodrigo Alonso", "Hsi-Ming Chang", "Elizabeth E. Jenkins", "Aneesh V. Manohar", "Brian Shotwell" ], "url": "https://arxiv.org/abs/1405.0486", "kind": "paper", "locatorHint": "Eq. (1) and Table I: four baryon-violating field structures and 273 complex coefficients in the three-generation ΔB = 1 sector without sterile neutrinos.", "licenseNote": "Linked primary research paper. Published flavor polynomials are independently recomputed; no upstream text or figures are redistributed." }, { "id": "kobach-2016-dimension-selection", "title": "Baryon Number, Lepton Number, and Operator Dimension in the Standard Model", "authors": [ "Andrew Kobach" ], "url": "https://arxiv.org/abs/1604.05726", "kind": "paper", "locatorHint": "Eqs. (15)–(18): d ≡ (ΔB − ΔL)/2 modulo 2 for Standard Model Lorentz- and hypercharge-invariant operators.", "licenseNote": "Linked primary research paper. 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It is simply connected and noncompact.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-c#physics-1", "subject": "group:sl-2-c", "key": "physics-1", "label": "In physics", "value": "Weyl and Dirac fields, Lorentz covariance, and the distinction between fields and states.", "evidence": [ { "source": "zuber", "locator": "Applications to the indicated symmetry and representation category" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-c#real-algebra-1", "subject": "group:sl-2-c", "key": "real-algebra-1", "label": "The Lorentz algebra is a realification", "value": "SL(2,C) is being used as a real six-dimensional Lie group. Its real Lie algebra complexifies to sl(2,C) ⊕ sl(2,C), rather than to a single A₁ factor. Finite-dimensional complex irreducibles consequently carry two nonnegative half-integer labels (jL,jR), with dimension (2jL+1)(2jR+1).", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "knapp-structure", "locator": "Theorem 6.105: sl(2,C) viewed as a real simple algebra" }, { "source": "etingof-lie-notes", "locator": "Section 9.4: real forms and complexification" } ], "derivation": { "method": "Complexify the realification, then take tensor products of irreducible modules for the two sl(2,C) factors. The simply connected real group integrates every finite-dimensional module.", "inputs": [ "algebra:sl-2-c-as-real", "algebra:a-1-plus-a-1-c", "SU(2) highest weight 2j has dimension 2j+1", "SL(2,C) is simply connected by polar decomposition" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-c#relation-1", "subject": "group:sl-2-c", "key": "relation-1", "label": "Related group", "value": "group:su-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "relationship": "related", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-c#relation-2", "subject": "group:sl-2-c", "key": "relation-2", "label": "Related group", "value": "group:spin-4", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "relationship": "related", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-c#relation-3", "subject": "group:sl-2-c", "key": "relation-3", "label": "Actual real Lie algebra", "value": "algebra:sl-2-c-as-real", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-c#relation-4", "subject": "group:sl-2-c", "key": "relation-4", "label": "Real-form context", "value": "real-form:sl-2-c-as-real", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-c#relation-5", "subject": "group:sl-2-c", "key": "relation-5", "label": "Explicit homomorphism", "value": "map:sl-2-c-to-lorentz", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#definition", "subject": "group:c-3", "key": "definition", "label": "Definition", "value": "⟨r | r³ = 1⟩", "evidence": [ { "source": "groupnames-c-3", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#order", "subject": "group:c-3", "key": "order", "label": "Order", "value": "3", "evidence": [ { "source": "groupnames-c-3", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#center", "subject": "group:c-3", "key": "center", "label": "Center", "value": "C₃", "evidence": [ { "source": "groupnames-c-3", "locator": "Abelian label; upper central series" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#abelian", "subject": "group:c-3", "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "groupnames-c-3", "locator": "Group-property labels; Commutator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#simple", "subject": "group:c-3", "key": "simple", "label": "Simple finite group", "value": "Yes", "evidence": [ { "source": "groupnames-c-3", "locator": "Simple label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#irreps", "subject": "group:c-3", "key": "irreps", "label": "Ordinary complex irrep dimensions", "value": "1, 1, 1", "evidence": [ { "source": "groupnames-c-3", "locator": "Character table: identity column and orthogonal/complex/symplectic types" }, { "source": "gap", "locator": "Irr; section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#representations-1", "subject": "group:c-3", "key": "representations-1", "label": "Representations", "value": "The generator acts by 1, ω, or ω², with ω = exp(2πi/3). 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Aliases; Generators and relations" } ], "assumptions": [], "relationship": "related", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#relation-3", "subject": "group:c-3", "key": "relation-3", "label": "Cyclic family", "value": "family:finite-cyclic", "evidence": [ { "source": "groupnames-c-3", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-3#relation-4", "subject": "group:c-3", "key": "relation-4", "label": "Prime-order simple family", "value": "family:finite-prime-cyclic", "evidence": [ { "source": "groupnames-c-3", "locator": "Simple label" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:s-3#definition", "subject": "group:s-3", "key": "definition", "label": "Definition", "value": "⟨r,s | r³ = s² = 1, srs = r⁻¹⟩", "evidence": [ { "source": "groupnames-s-3", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:s-3#order", "subject": "group:s-3", "key": "order", "label": "Order", "value": "6", "evidence": [ { "source": "groupnames-s-3", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:s-3#center", "subject": "group:s-3", "key": "center", "label": "Center", "value": "Trivial", "evidence": [ { "source": "groupnames-s-3", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:s-3#abelian", "subject": "group:s-3", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "groupnames-s-3", "locator": "Group-property labels; Commutator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:s-3#simple", "subject": "group:s-3", "key": "simple", "label": "Simple finite group", "value": "No", "evidence": [ { "source": "groupnames-s-3", "locator": "Chief series and normal subgroups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:s-3#irreps", "subject": "group:s-3", "key": "irreps", "label": "Ordinary complex irrep dimensions", "value": "1, 1, 2", "evidence": [ { "source": "groupnames-s-3", "locator": "Character table: identity column and orthogonal/complex/symplectic types" }, { "source": "gap", "locator": "Irr; 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Aliases; Generators and relations" } ], "assumptions": [], "relationship": "related", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-5#relation-2", "subject": "group:a-5", "key": "relation-2", "label": "Related group", "value": "group:so-3", "evidence": [ { "source": "groupnames-a-5", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "relationship": "related", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-5#relation-3", "subject": "group:a-5", "key": "relation-3", "label": "Alternating simple family", "value": "family:finite-alternating", "evidence": [ { "source": "groupnames-a-5", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-5#relation-4", "subject": "group:a-5", "key": "relation-4", "label": "PSL(2,4) ≅ PSL(2,5)", "value": "family:finite-linear", "evidence": [ { "source": "groupnames-a-5", "locator": "Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-8#definition", "subject": "group:q-8", "key": "definition", "label": "Definition", "value": "i² = j² = k² = ijk = −1", "evidence": [ { "source": "groupnames-q-8", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-8#order", "subject": "group:q-8", "key": "order", "label": "Order", "value": "8", "evidence": [ { "source": "groupnames-q-8", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-8#center", "subject": "group:q-8", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "groupnames-q-8", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-8#abelian", "subject": "group:q-8", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "groupnames-q-8", "locator": "Group-property labels; Commutator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-8#simple", "subject": "group:q-8", "key": "simple", "label": "Simple finite group", "value": "No", "evidence": [ { "source": "groupnames-q-8", "locator": "Chief series and normal subgroups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-8#irreps", "subject": "group:q-8", "key": "irreps", "label": "Ordinary complex irrep dimensions", "value": "1, 1, 1, 1, 2", "evidence": [ { "source": "groupnames-q-8", "locator": "Character table: identity column and orthogonal/complex/symplectic types" }, { "source": "gap", "locator": "Irr; section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-8#representations-1", "subject": "group:q-8", "key": "representations-1", "label": "Representations", "value": "Four real-type one-dimensional irreps and one quaternionic-type doublet. 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Its irrep dimensions match Q₈, but the doublet has a different reality type.", "evidence": [ { "source": "groupnames-d-8", "locator": "Character table; matrix and permutation representations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-8#physics-1", "subject": "group:d-8", "key": "physics-1", "label": "In physics", "value": "Point-group actions, selection rules, and comparisons of groups with similar character data.", "evidence": [ { "source": "groupnames-d-8", "locator": "Character table; matrix and permutation representations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-8#relation-1", "subject": "group:d-8", "key": "relation-1", "label": "Related group", "value": "group:q-8", "evidence": [ { "source": "groupnames-d-8", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "relationship": "related", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-8#relation-2", "subject": "group:d-8", "key": "relation-2", "label": "Related group", "value": "group:s-3", "evidence": [ { "source": "groupnames-d-8", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "relationship": "related", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-8#relation-3", "subject": "group:d-8", "key": "relation-3", "label": "Dihedral family; order eight", "value": "family:finite-dihedral", "evidence": [ { "source": "groupnames-d-8", "locator": "Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-prime-cyclic#definition", "subject": "family:finite-prime-cyclic", "key": "definition", "label": "Definition", "value": "The additive group of the prime field Fₚ, equivalently a cyclic group generated by an element of order p.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Simple groups, first slide" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-prime-cyclic#parameters", "subject": "family:finite-prime-cyclic", "key": "parameters", "label": "Parameter domain", "value": "p: prime number", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Simple groups, first slide" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-prime-cyclic#restriction", "subject": "family:finite-prime-cyclic", "key": "restriction", "label": "Scope and restrictions", "value": "Exactly the abelian finite simple groups.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Simple groups, first slide" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-prime-cyclic#order-formula", "subject": "family:finite-prime-cyclic", "key": "order-formula", "label": "Order", "value": "p", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-prime-cyclic#conventions-1", "subject": "family:finite-prime-cyclic", "key": "conventions-1", "label": "Conventions and use", "value": "The trivial group is excluded from the definition of a simple group.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Simple groups, first slide" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-prime-cyclic#relation-1", "subject": "family:finite-prime-cyclic", "key": "relation-1", "label": "Prime-order cyclic subfamily", "value": "family:finite-cyclic", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [], "relationship": "subfamily-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#definition", "subject": "family:finite-alternating", "key": "definition", "label": "Definition", "value": "The subgroup of even permutations in the symmetric group Sₙ.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG; even permutations; simplicity of Aₙ" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#parameters", "subject": "family:finite-alternating", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 5", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG; even permutations; simplicity of Aₙ" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#restriction", "subject": "family:finite-alternating", "key": "restriction", "label": "Scope and restrictions", "value": "Aₙ is nonabelian simple for n ≥ 5.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG; even permutations; simplicity of Aₙ" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#order-formula", "subject": "family:finite-alternating", "key": "order-formula", "label": "Order", "value": "n! / 2", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Even permutations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#exceptions-1", "subject": "family:finite-alternating", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "n = 5: A₅ ≅ PSL(2,4) ≅ PSL(2,5).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#exceptions-2", "subject": "family:finite-alternating", "key": "exceptions-2", "label": "Exceptions and identifications", "value": "n = 6: A₆ ≅ PSL(2,9) ≅ PSp(4,2)′.", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#exceptions-3", "subject": "family:finite-alternating", "key": "exceptions-3", "label": "Exceptions and identifications", "value": "n = 8: A₈ ≅ PSL(4,2).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#relation-1", "subject": "family:finite-alternating", "key": "relation-1", "label": "A₅ reference", "value": "group:a-5", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-alternating#relation-2", "subject": "family:finite-alternating", "key": "relation-2", "label": "A₆ reference", "value": "group:a-6", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#definition", "subject": "family:finite-linear", "key": "definition", "label": "Definition", "value": "SL(n,Fq) modulo its scalar center; the untwisted finite group of Lie type Aₙ₋₁.", "evidence": [ { "source": "finite-wilson-linear", "locator": "Definitions; more examples; simplicity of PSLₙ(q)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#parameters", "subject": "family:finite-linear", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 2; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-linear", "locator": "Definitions; more examples; simplicity of PSLₙ(q)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#restriction", "subject": "family:finite-linear", "key": "restriction", "label": "Scope and restrictions", "value": "Exclude (n,q) = (2,2) and (2,3).", "evidence": [ { "source": "finite-wilson-linear", "locator": "Definitions; more examples; simplicity of PSLₙ(q)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#order-formula", "subject": "family:finite-linear", "key": "order-formula", "label": "Order", "value": "q^[n(n−1)/2] · ∏ᵢ₌₂ⁿ(qⁱ−1) / gcd(n,q−1)", "evidence": [ { "source": "finite-wilson-linear", "locator": "Orders of linear groups and scalar quotient" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#exceptions-1", "subject": "family:finite-linear", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "(2,2), (2,3): PSL(2,2) ≅ S₃ and PSL(2,3) ≅ A₄ are not simple.", "evidence": [ { "source": "finite-wilson-linear", "locator": "Definitions; more examples; simplicity of PSLₙ(q)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#exceptions-2", "subject": "family:finite-linear", "key": "exceptions-2", "label": "Exceptions and identifications", "value": "(2,4), (2,5): Both identify with A₅.", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#exceptions-3", "subject": "family:finite-linear", "key": "exceptions-3", "label": "Exceptions and identifications", "value": "(2,7), (3,2): PSL(2,7) ≅ PSL(3,2).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#exceptions-4", "subject": "family:finite-linear", "key": "exceptions-4", "label": "Exceptions and identifications", "value": "(2,9): PSL(2,9) ≅ A₆.", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#exceptions-5", "subject": "family:finite-linear", "key": "exceptions-5", "label": "Exceptions and identifications", "value": "(4,2): PSL(4,2) ≅ A₈.", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#relation-1", "subject": "family:finite-linear", "key": "relation-1", "label": "Canonical A₅ record", "value": "group:a-5", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#relation-2", "subject": "family:finite-linear", "key": "relation-2", "label": "Canonical PSL(2,7) record", "value": "group:psl-2-7", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-linear#relation-3", "subject": "family:finite-linear", "key": "relation-3", "label": "Canonical A₆ record", "value": "group:a-6", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-unitary#definition", "subject": "family:finite-unitary", "key": "definition", "label": "Definition", "value": "The determinant-one isometries of a nondegenerate Hermitian form over F(q²), modulo scalar matrices.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Unitary groups; structure of unitary groups I–II" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-unitary#parameters", "subject": "family:finite-unitary", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 3; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-classical", "locator": "Unitary groups; structure of unitary groups I–II" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-unitary#restriction", "subject": "family:finite-unitary", "key": "restriction", "label": "Scope and restrictions", "value": "Exclude (n,q) = (3,2); q is the fixed-field size, not the matrix-field size.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Unitary groups; structure of unitary groups I–II" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-unitary#order-formula", "subject": "family:finite-unitary", "key": "order-formula", "label": "Order", "value": "q^[n(n−1)/2] · ∏ᵢ₌₂ⁿ(qⁱ−(−1)ⁱ) / gcd(n,q+1)", "evidence": [ { "source": "finite-wilson-classical", "locator": "Orders and scalar center of unitary groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-unitary#exceptions-1", "subject": "family:finite-unitary", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "n = 2: PSU(2,q) ≅ PSL(2,q); these are indexed in the linear family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Unitary groups; structure of unitary groups I–II" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-unitary#exceptions-2", "subject": "family:finite-unitary", "key": "exceptions-2", "label": "Exceptions and identifications", "value": "(3,2): PSU(3,2) has order 72 and is not simple.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Unitary groups; structure of unitary groups I–II" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-unitary#exceptions-3", "subject": "family:finite-unitary", "key": "exceptions-3", "label": "Exceptions and identifications", "value": "(4,2): PSU(4,2) ≅ PSp(4,3).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symplectic#definition", "subject": "family:finite-symplectic", "key": "definition", "label": "Definition", "value": "The isometries of a nondegenerate alternating form on Fq^(2n), modulo scalar matrices.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.2.1–3.2.2" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symplectic#parameters", "subject": "family:finite-symplectic", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 2; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.2.1–3.2.2" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symplectic#restriction", "subject": "family:finite-symplectic", "key": "restriction", "label": "Scope and restrictions", "value": "Exclude (n,q) = (2,2).", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.2.1–3.2.2" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symplectic#order-formula", "subject": "family:finite-symplectic", "key": "order-formula", "label": "Order", "value": "q^(n²) · ∏ᵢ₌₁ⁿ(q^(2i)−1) / gcd(2,q−1)", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.2.1, order and scalar quotient" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symplectic#exceptions-1", "subject": "family:finite-symplectic", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "n = 1: PSp(2,q) ≅ PSL(2,q); use the linear family.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.2.1–3.2.2" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symplectic#exceptions-2", "subject": "family:finite-symplectic", "key": "exceptions-2", "label": "Exceptions and identifications", "value": "(2,2): Sp(4,2) ≅ S₆; its derived subgroup is A₆.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.2.2" }, { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symplectic#exceptions-3", "subject": "family:finite-symplectic", "key": "exceptions-3", "label": "Exceptions and identifications", "value": "(2,3): PSp(4,3) ≅ PSU(4,2).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symplectic#relation-1", "subject": "family:finite-symplectic", "key": "relation-1", "label": "A₆ as the derived small case", "value": "group:a-6", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-odd-orthogonal#definition", "subject": "family:finite-odd-orthogonal", "key": "definition", "label": "Definition", "value": "The projective Ω group of a nondegenerate quadratic form in odd dimension over Fq.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.4, orthogonal groups in odd characteristic" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-odd-orthogonal#parameters", "subject": "family:finite-odd-orthogonal", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 3; q: odd prime power", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.4, orthogonal groups in odd characteristic" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-odd-orthogonal#restriction", "subject": "family:finite-odd-orthogonal", "key": "restriction", "label": "Scope and restrictions", "value": "Use odd q and n ≥ 3 to avoid the symplectic and low-rank repetitions.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.4, orthogonal groups in odd characteristic" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-odd-orthogonal#exceptions-1", "subject": "family:finite-odd-orthogonal", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "dimension 3: PΩ(3,q) ≅ PSL(2,q).", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup: B-type conventions" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-odd-orthogonal#exceptions-2", "subject": "family:finite-odd-orthogonal", "key": "exceptions-2", "label": "Exceptions and identifications", "value": "dimension 5: PΩ(5,q) ≅ PSp(4,q).", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-odd-orthogonal#exceptions-3", "subject": "family:finite-odd-orthogonal", "key": "exceptions-3", "label": "Exceptions and identifications", "value": "even q: The odd-dimensional orthogonal groups identify with the corresponding symplectic groups.", "evidence": [ { "source": "finite-gap-classical-maximals", "locator": "Section 2.2-4, first paragraph" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-odd-orthogonal#conventions-1", "subject": "family:finite-odd-orthogonal", "key": "conventions-1", "label": "Conventions and use", "value": "Ω denotes the subgroup required for simplicity; merely taking a projective special orthogonal group does not in general give the simple group.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Section 3.4, orthogonal groups in odd characteristic" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-plus#definition", "subject": "family:finite-even-orthogonal-plus", "key": "definition", "label": "Definition", "value": "The projective Ω group of a nondegenerate split quadratic form on Fq^(2n).", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, plus type and projective Ω groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-plus#parameters", "subject": "family:finite-even-orthogonal-plus", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 4; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, plus type and projective Ω groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-plus#restriction", "subject": "family:finite-even-orthogonal-plus", "key": "restriction", "label": "Scope and restrictions", "value": "n ≥ 4; plus type has Witt index n.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, plus type and projective Ω groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-plus#exceptions-1", "subject": "family:finite-even-orthogonal-plus", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "dimension 4: PΩ⁺(4,q) ≅ PSL(2,q) × PSL(2,q), so this is not a new simple family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-plus#exceptions-2", "subject": "family:finite-even-orthogonal-plus", "key": "exceptions-2", "label": "Exceptions and identifications", "value": "dimension 6: PΩ⁺(6,q) ≅ PSL(4,q); use the linear family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-minus#definition", "subject": "family:finite-even-orthogonal-minus", "key": "definition", "label": "Definition", "value": "The projective Ω group of a nonsplit nondegenerate quadratic form on Fq^(2n).", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, minus type and projective Ω groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-minus#parameters", "subject": "family:finite-even-orthogonal-minus", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 4; q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, minus type and projective Ω groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-minus#restriction", "subject": "family:finite-even-orthogonal-minus", "key": "restriction", "label": "Scope and restrictions", "value": "n ≥ 4; minus type has Witt index n−1.", "evidence": [ { "source": "finite-wilson-classical-notes", "locator": "Sections 3.4–3.5, minus type and projective Ω groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-minus#exceptions-1", "subject": "family:finite-even-orthogonal-minus", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "dimension 4: PΩ⁻(4,q) ≅ PSL(2,q²); use the linear family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-even-orthogonal-minus#exceptions-2", "subject": "family:finite-even-orthogonal-minus", "key": "exceptions-2", "label": "Exceptions and identifications", "value": "dimension 6: PΩ⁻(6,q) ≅ PSU(4,q); use the unitary family.", "evidence": [ { "source": "finite-wilson-classical", "locator": "Small-dimensional orthogonal groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-g2#definition", "subject": "family:finite-g2", "key": "definition", "label": "Definition", "value": "The untwisted Chevalley group of type G₂ over Fq.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "G₂(q), order and simplicity" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-g2#parameters", "subject": "family:finite-g2", "key": "parameters", "label": "Parameter domain", "value": "q: prime power q ≥ 3", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "G₂(q), order and simplicity" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-g2#restriction", "subject": "family:finite-g2", "key": "restriction", "label": "Scope and restrictions", "value": "q ≥ 3; the q = 2 parent group is not simple.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "G₂(q), order and simplicity" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-g2#exceptions-1", "subject": "family:finite-g2", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "q = 2: G₂(2)′ ≅ PSU(3,3), of index 2 in G₂(2).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" }, { "source": "finite-wilson-exceptional", "locator": "G₂(q), small case" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-f4#definition", "subject": "family:finite-f4", "key": "definition", "label": "Definition", "value": "The untwisted Chevalley finite group of Lie type F₄, in its simple central-quotient form.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-f4#parameters", "subject": "family:finite-f4", "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-f4#restriction", "subject": "family:finite-f4", "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-f4#conventions-1", "subject": "family:finite-f4", "key": "conventions-1", "label": "Conventions and use", "value": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-e6#definition", "subject": "family:finite-e6", "key": "definition", "label": "Definition", "value": "The untwisted Chevalley finite group of Lie type E₆, in its simple central-quotient form.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-e6#parameters", "subject": "family:finite-e6", "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-e6#restriction", "subject": "family:finite-e6", "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-e6#conventions-1", "subject": "family:finite-e6", "key": "conventions-1", "label": "Conventions and use", "value": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-e7#definition", "subject": "family:finite-e7", "key": "definition", "label": "Definition", "value": "The untwisted Chevalley finite group of Lie type E₇, in its simple central-quotient form.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-e7#parameters", "subject": "family:finite-e7", "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-e7#restriction", "subject": "family:finite-e7", "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; 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take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-e8#conventions-1", "subject": "family:finite-e8", "key": "conventions-1", "label": "Conventions and use", "value": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-twisted-e6#definition", "subject": "family:finite-twisted-e6", "key": "definition", "label": "Definition", "value": "The graph-field twisted Steinberg finite group of Lie type ²E₆, in its simple central-quotient form.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-twisted-e6#parameters", "subject": "family:finite-twisted-e6", "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-twisted-e6#restriction", "subject": "family:finite-twisted-e6", "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-twisted-e6#conventions-1", "subject": "family:finite-twisted-e6", "key": "conventions-1", "label": "Conventions and use", "value": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-triality#definition", "subject": "family:finite-triality", "key": "definition", "label": "Definition", "value": "The triality-twisted Steinberg finite group of Lie type ³D₄, in its simple central-quotient form.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-triality#parameters", "subject": "family:finite-triality", "key": "parameters", "label": "Parameter domain", "value": "q: q = p^f, with p prime and f ≥ 1", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-triality#restriction", "subject": "family:finite-triality", "key": "restriction", "label": "Scope and restrictions", "value": "All prime powers q ≥ 2; take the simple central quotient.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Ten exceptional families" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-triality#conventions-1", "subject": "family:finite-triality", "key": "conventions-1", "label": "Conventions and use", "value": "The finite-field object is distinct from a compact or complex Lie group with the same root-system label.", "evidence": [ { "source": "finite-gap-groups", "locator": "IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-suzuki#definition", "subject": "family:finite-suzuki", "key": "definition", "label": "Definition", "value": "The Suzuki twisted groups of type ²B₂.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Suzuki construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-suzuki#parameters", "subject": "family:finite-suzuki", "key": "parameters", "label": "Parameter domain", "value": "q: q = 2^(2m+1), integer m ≥ 1", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Suzuki construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-suzuki#restriction", "subject": "family:finite-suzuki", "key": "restriction", "label": "Scope and restrictions", "value": "Odd extension degree in characteristic 2; q ≥ 8.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Suzuki construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-suzuki#exceptions-1", "subject": "family:finite-suzuki", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "m = 0; q = 2: Sz(2) has order 20 and is not simple.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree groups: small cases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-suzuki#conventions-1", "subject": "family:finite-suzuki", "key": "conventions-1", "label": "Conventions and use", "value": "Sz(q) denotes this infinite family; Suz denotes the distinct sporadic Suzuki group.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Suzuki construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-suzuki#relation-1", "subject": "family:finite-suzuki", "key": "relation-1", "label": "Sporadic Suz: different group", "value": "group:sporadic-suz", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Suz entry" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-g2#definition", "subject": "family:finite-ree-g2", "key": "definition", "label": "Definition", "value": "The Ree twisted groups of type ²G₂.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Small Ree construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-g2#parameters", "subject": "family:finite-ree-g2", "key": "parameters", "label": "Parameter domain", "value": "q: q = 3^(2m+1), integer m ≥ 1", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Small Ree construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-g2#restriction", "subject": "family:finite-ree-g2", "key": "restriction", "label": "Scope and restrictions", "value": "Odd extension degree in characteristic 3; q ≥ 27.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Small Ree construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-g2#exceptions-1", "subject": "family:finite-ree-g2", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "m = 0; q = 3: ²G₂(3)′ ≅ PSL(2,8), with index 3 in ²G₂(3).", "evidence": [ { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" }, { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree groups: small cases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-f4#definition", "subject": "family:finite-ree-f4", "key": "definition", "label": "Definition", "value": "The Ree twisted groups of type ²F₄.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Large Ree construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-f4#parameters", "subject": "family:finite-ree-f4", "key": "parameters", "label": "Parameter domain", "value": "q: q = 2^(2m+1), integer m ≥ 1", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Large Ree construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-f4#restriction", "subject": "family:finite-ree-f4", "key": "restriction", "label": "Scope and restrictions", "value": "Odd extension degree in characteristic 2; q ≥ 8. The Tits group is a separate small-case entry.", "evidence": [ { "source": "finite-wilson-suzuki-ree", "locator": "Large Ree construction and simplicity; section 5" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-f4#exceptions-1", "subject": "family:finite-ree-f4", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "m = 0; q = 2: ²F₄(2) is not simple. Its index-two derived subgroup ²F₄(2)′ is the Tits simple group.", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree groups: small cases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-ree-f4#relation-1", "subject": "family:finite-ree-f4", "key": "relation-1", "label": "Tits derived subgroup", "value": "group:tits", "evidence": [ { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree groups: small cases" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-sporadic#definition", "subject": "family:finite-sporadic", "key": "definition", "label": "Definition", "value": "The finite list of nonabelian simple groups outside the alternating and Lie-type families.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-sporadic#parameters", "subject": "family:finite-sporadic", "key": "parameters", "label": "Parameter domain", "value": "Finite collection", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-sporadic#restriction", "subject": "family:finite-sporadic", "key": "restriction", "label": "Scope and restrictions", "value": "Exactly 26 exceptional finite simple groups; the Tits group is excluded from this count.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-sporadic#exceptions-1", "subject": "family:finite-sporadic", "key": "exceptions-1", "label": "Exceptions and identifications", "value": "Tits convention: ²F₄(2)′ is indexed with the Lie-type small cases, not as a 27th sporadic group.", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Miscellaneous: Tits group" }, { "source": "finite-wilson-exceptional", "locator": "Suzuki–Ree small cases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-sporadic#conventions-1", "subject": "family:finite-sporadic", "key": "conventions-1", "label": "Conventions and use", "value": "Individual entries record identity and order. The linked ATLAS separates representations by group, field and characteristic.", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "Sporadic index and individual group pages" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-cyclic#definition", "subject": "family:finite-cyclic", "key": "definition", "label": "Definition", "value": "The group generated by a with relation aⁿ = 1, with a of exact order n.", "evidence": [ { "source": "finite-gap-constructors", "locator": "CyclicGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-cyclic#parameters", "subject": "family:finite-cyclic", "key": "parameters", "label": "Parameter domain", "value": "n: positive integer", "evidence": [ { "source": "finite-gap-constructors", "locator": "CyclicGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-cyclic#restriction", "subject": "family:finite-cyclic", "key": "restriction", "label": "Scope and restrictions", "value": "One cyclic group for every finite order n.", "evidence": [ { "source": "finite-gap-constructors", "locator": "CyclicGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-cyclic#conventions-1", "subject": "family:finite-cyclic", "key": "conventions-1", "label": "Conventions and use", "value": "Ordinary complex irreducibles are the one-dimensional characters χₖ(a) = exp(2πik/n), with k taken modulo n.", "evidence": [ { "source": "gap", "locator": "Ordinary character tables and irreducible characters" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-dihedral#definition", "subject": "family:finite-dihedral", "key": "definition", "label": "Definition", "value": "Symmetries of a regular n-gon: rⁿ = s² = 1 and srs⁻¹ = r⁻¹.", "evidence": [ { "source": "finite-gap-constructors", "locator": "DihedralGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-dihedral#parameters", "subject": "family:finite-dihedral", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 3", "evidence": [ { "source": "finite-gap-constructors", "locator": "DihedralGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-dihedral#restriction", "subject": "family:finite-dihedral", "key": "restriction", "label": "Scope and restrictions", "value": "This site writes D₂ₙ for the group of order 2n.", "evidence": [ { "source": "finite-gap-constructors", "locator": "DihedralGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-dihedral#conventions-1", "subject": "family:finite-dihedral", "key": "conventions-1", "label": "Conventions and use", "value": "A competing convention calls the same group Dₙ. Check the order whenever importing a dihedral label.", "evidence": [ { "source": "finite-gap-constructors", "locator": "DihedralGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-dicyclic#definition", "subject": "family:finite-dicyclic", "key": "definition", "label": "Definition", "value": "The group with a^(2n) = 1, x² = aⁿ and xax⁻¹ = a⁻¹.", "evidence": [ { "source": "finite-gap-constructors", "locator": "Sections 50.1-8 DicyclicGroup and 50.1-9 IsDicyclicGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-dicyclic#parameters", "subject": "family:finite-dicyclic", "key": "parameters", "label": "Parameter domain", "value": "n: integer n ≥ 2", "evidence": [ { "source": "finite-gap-constructors", "locator": "Sections 50.1-8 DicyclicGroup and 50.1-9 IsDicyclicGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-dicyclic#restriction", "subject": "family:finite-dicyclic", "key": "restriction", "label": "Scope and restrictions", "value": "Dicₙ has order 4n; Dic₂ = Q₈ and Dic₄ = Q₁₆.", "evidence": [ { "source": "finite-gap-constructors", "locator": "Sections 50.1-8 DicyclicGroup and 50.1-9 IsDicyclicGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-dicyclic#conventions-1", "subject": "family:finite-dicyclic", "key": "conventions-1", "label": "Conventions and use", "value": "For n a power of two this is a generalized quaternion group.", "evidence": [ { "source": "finite-gap-constructors", "locator": "Sections 50.1-8 DicyclicGroup and 50.1-9 IsDicyclicGroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:binary-polyhedral#definition", "subject": "family:binary-polyhedral", "key": "definition", "label": "Definition", "value": "The inverse images in SU(2) of the tetrahedral, octahedral and icosahedral rotation groups under SU(2) → SO(3).", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups, slides 7–8" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:binary-polyhedral#parameters", "subject": "family:binary-polyhedral", "key": "parameters", "label": "Parameter domain", "value": "G: 2T, 2O or 2I", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups, slides 7–8" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:binary-polyhedral#restriction", "subject": "family:binary-polyhedral", "key": "restriction", "label": "Scope and restrictions", "value": "The binary tetrahedral, octahedral and icosahedral groups.", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups, slides 7–8" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:binary-polyhedral#relation-1", "subject": "family:binary-polyhedral", "key": "relation-1", "label": "Finite subgroups of SU(2)", "value": "group:su-2", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups, slides 7–8" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases: binary polyhedral group; Centre and rotational quotient" } ], "assumptions": [], "relationship": "related-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symmetric#definition", "subject": "family:finite-symmetric", "key": "definition", "label": "Definition", "value": "The group of bijections of an n-element set, with composition as the operation.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Permutations; even permutations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symmetric#parameters", "subject": "family:finite-symmetric", "key": "parameters", "label": "Parameter domain", "value": "n: positive integer", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Permutations; even permutations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symmetric#restriction", "subject": "family:finite-symmetric", "key": "restriction", "label": "Scope and restrictions", "value": "All permutations of n symbols; order n!.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Permutations; even permutations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "family:finite-symmetric#conventions-1", "subject": "family:finite-symmetric", "key": "conventions-1", "label": "Conventions and use", "value": "The sign homomorphism has kernel Aₙ; for n ≥ 5 that kernel is nonabelian simple.", "evidence": [ { "source": "finite-wilson-introduction", "locator": "Even permutations; simplicity of alternating groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#definition", "subject": "group:c-2", "key": "definition", "label": "Definition", "value": "The cyclic group generated by a with a^2 = 1 and exact order 2.", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#order", "subject": "group:c-2", "key": "order", "label": "Order", "value": "2", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#center", "subject": "group:c-2", "key": "center", "label": "Center", "value": "C2", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Abelian label; upper central series" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#abelian", "subject": "group:c-2", "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Abelian label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#simple", "subject": "group:c-2", "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Simple label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#irreps", "subject": "group:c-2", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#smallgroups-id", "subject": "group:c-2", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(2,1)", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#representations-1", "subject": "group:c-2", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#representations-2", "subject": "group:c-2", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#structure-1", "subject": "group:c-2", "key": "structure-1", "label": "Structure and conventions", "value": "For a generator a, χₖ(a) = exp(2πik/2); source row labels need not be ordered by k.", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#relation-1", "subject": "group:c-2", "key": "relation-1", "label": "Family", "value": "family:finite-cyclic", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-2#relation-2", "subject": "group:c-2", "key": "relation-2", "label": "Family", "value": "family:finite-prime-cyclic", "evidence": [ { "source": "finite-groupnames-c-2", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#definition", "subject": "group:c-4", "key": "definition", "label": "Definition", "value": "The cyclic group generated by a with a^4 = 1 and exact order 4.", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#order", "subject": "group:c-4", "key": "order", "label": "Order", "value": "4", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#center", "subject": "group:c-4", "key": "center", "label": "Center", "value": "C4", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Abelian label; upper central series" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#abelian", "subject": "group:c-4", "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Abelian label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#simple", "subject": "group:c-4", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#irreps", "subject": "group:c-4", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#smallgroups-id", "subject": "group:c-4", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(4,1)", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#representations-1", "subject": "group:c-4", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. 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The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#structure-1", "subject": "group:c-4", "key": "structure-1", "label": "Structure and conventions", "value": "For a generator a, χₖ(a) = exp(2πik/4); source row labels need not be ordered by k.", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-4#relation-1", "subject": "group:c-4", "key": "relation-1", "label": "Family", "value": "family:finite-cyclic", "evidence": [ { "source": "finite-groupnames-c-4", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#definition", "subject": "group:c-5", "key": "definition", "label": "Definition", "value": "The cyclic group generated by a with a^5 = 1 and exact order 5.", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#order", "subject": "group:c-5", "key": "order", "label": "Order", "value": "5", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#center", "subject": "group:c-5", "key": "center", "label": "Center", "value": "C5", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Abelian label; upper central series" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#abelian", "subject": "group:c-5", "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Abelian label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#simple", "subject": "group:c-5", "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Simple label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#irreps", "subject": "group:c-5", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 1", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#smallgroups-id", "subject": "group:c-5", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(5,1)", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#representations-1", "subject": "group:c-5", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. 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The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#structure-1", "subject": "group:c-5", "key": "structure-1", "label": "Structure and conventions", "value": "For a generator a, χₖ(a) = exp(2πik/5); source row labels need not be ordered by k.", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#relation-1", "subject": "group:c-5", "key": "relation-1", "label": "Family", "value": "family:finite-cyclic", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-5#relation-2", "subject": "group:c-5", "key": "relation-2", "label": "Family", "value": "family:finite-prime-cyclic", "evidence": [ { "source": "finite-groupnames-c-5", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#definition", "subject": "group:c-6", "key": "definition", "label": "Definition", "value": "The cyclic group generated by a with a^6 = 1 and exact order 6.", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#order", "subject": "group:c-6", "key": "order", "label": "Order", "value": "6", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#center", "subject": "group:c-6", "key": "center", "label": "Center", "value": "C6", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Abelian label; upper central series" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#abelian", "subject": "group:c-6", "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Abelian label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#simple", "subject": "group:c-6", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#irreps", "subject": "group:c-6", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 1, 1", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#smallgroups-id", "subject": "group:c-6", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(6,2)", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#representations-1", "subject": "group:c-6", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#representations-2", "subject": "group:c-6", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#structure-1", "subject": "group:c-6", "key": "structure-1", "label": "Structure and conventions", "value": "For a generator a, χₖ(a) = exp(2πik/6); source row labels need not be ordered by k.", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:c-6#relation-1", "subject": "group:c-6", "key": "relation-1", "label": "Family", "value": "family:finite-cyclic", "evidence": [ { "source": "finite-groupnames-c-6", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#definition", "subject": "group:v-4", "key": "definition", "label": "Definition", "value": "C₂ × C₂, generated by two commuting involutions.", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#order", "subject": "group:v-4", "key": "order", "label": "Order", "value": "4", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#center", "subject": "group:v-4", "key": "center", "label": "Center", "value": "V₄", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Abelian label; upper central series" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#abelian", "subject": "group:v-4", "key": "abelian", "label": "Abelian", "value": "Yes", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Abelian label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#simple", "subject": "group:v-4", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#irreps", "subject": "group:v-4", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#smallgroups-id", "subject": "group:v-4", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(4,2)", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#representations-1", "subject": "group:v-4", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#representations-2", "subject": "group:v-4", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#structure-1", "subject": "group:v-4", "key": "structure-1", "label": "Structure and conventions", "value": "Every nonidentity element has order two.", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:v-4#relation-1", "subject": "group:v-4", "key": "relation-1", "label": "Normal Klein subgroup of A₄", "value": "group:a-4", "evidence": [ { "source": "finite-groupnames-v-4", "locator": "Subgroup lattice and maximal subgroup list" } ], "assumptions": [], "relationship": "normal-subgroup-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#definition", "subject": "group:d-10", "key": "definition", "label": "Definition", "value": "Symmetries of the regular pentagon: r⁵ = s² = 1 and srs⁻¹ = r⁻¹.", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#order", "subject": "group:d-10", "key": "order", "label": "Order", "value": "10", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#center", "subject": "group:d-10", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#abelian", "subject": "group:d-10", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#simple", "subject": "group:d-10", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#irreps", "subject": "group:d-10", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 2, 2", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#smallgroups-id", "subject": "group:d-10", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(10,1)", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#representations-1", "subject": "group:d-10", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#representations-2", "subject": "group:d-10", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#structure-1", "subject": "group:d-10", "key": "structure-1", "label": "Structure and conventions", "value": "D₁₀ here means order ten. The cited GroupNames page uses D₅, with the subscript counting polygon vertices.", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-10#relation-1", "subject": "group:d-10", "key": "relation-1", "label": "Family", "value": "family:finite-dihedral", "evidence": [ { "source": "finite-groupnames-d-10", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#definition", "subject": "group:d-12", "key": "definition", "label": "Definition", "value": "Symmetries of the regular hexagon: r⁶ = s² = 1 and srs⁻¹ = r⁻¹.", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#order", "subject": "group:d-12", "key": "order", "label": "Order", "value": "12", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#center", "subject": "group:d-12", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#abelian", "subject": "group:d-12", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#simple", "subject": "group:d-12", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#irreps", "subject": "group:d-12", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 2, 2", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#smallgroups-id", "subject": "group:d-12", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(12,4)", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#representations-1", "subject": "group:d-12", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#representations-2", "subject": "group:d-12", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#structure-1", "subject": "group:d-12", "key": "structure-1", "label": "Structure and conventions", "value": "The central half-turn generates C₂. Quotienting by the center gives S₃. The cited page calls this group D₆.", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#relation-1", "subject": "group:d-12", "key": "relation-1", "label": "Family", "value": "family:finite-dihedral", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:d-12#relation-2", "subject": "group:d-12", "key": "relation-2", "label": "D₁₂ / C₂ ≅ S₃", "value": "group:s-3", "evidence": [ { "source": "finite-groupnames-d-12", "locator": "Centre: G/Z" } ], "assumptions": [], "relationship": "central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#definition", "subject": "group:dic-3", "key": "definition", "label": "Definition", "value": "The group with a⁶ = 1, x² = a³ and xax⁻¹ = a⁻¹.", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#order", "subject": "group:dic-3", "key": "order", "label": "Order", "value": "12", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#center", "subject": "group:dic-3", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#abelian", "subject": "group:dic-3", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#simple", "subject": "group:dic-3", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#irreps", "subject": "group:dic-3", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 2, 2", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#smallgroups-id", "subject": "group:dic-3", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(12,1)", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#representations-1", "subject": "group:dic-3", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#representations-2", "subject": "group:dic-3", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#structure-1", "subject": "group:dic-3", "key": "structure-1", "label": "Structure and conventions", "value": "Its abelianization is C₄; two singlets form a complex-conjugate pair.", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#relation-1", "subject": "group:dic-3", "key": "relation-1", "label": "Family", "value": "family:finite-dicyclic", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:dic-3#relation-2", "subject": "group:dic-3", "key": "relation-2", "label": "Dic₃ / C₂ ≅ S₃", "value": "group:s-3", "evidence": [ { "source": "finite-groupnames-dic-3", "locator": "Centre: G/Z" } ], "assumptions": [], "relationship": "central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#definition", "subject": "group:q-16", "key": "definition", "label": "Definition", "value": "The group with a⁸ = 1, x² = a⁴ and xax⁻¹ = a⁻¹.", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#order", "subject": "group:q-16", "key": "order", "label": "Order", "value": "16", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#center", "subject": "group:q-16", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#abelian", "subject": "group:q-16", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#simple", "subject": "group:q-16", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#irreps", "subject": "group:q-16", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 2, 2, 2", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#smallgroups-id", "subject": "group:q-16", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(16,9)", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#representations-1", "subject": "group:q-16", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#representations-2", "subject": "group:q-16", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#relation-1", "subject": "group:q-16", "key": "relation-1", "label": "Family", "value": "family:finite-dicyclic", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:q-16#relation-2", "subject": "group:q-16", "key": "relation-2", "label": "Q₁₆ / C₂ ≅ D₈", "value": "group:d-8", "evidence": [ { "source": "finite-groupnames-q-16", "locator": "Centre: quotient D₄ in the order-eight convention" } ], "assumptions": [], "relationship": "central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#definition", "subject": "group:binary-tetrahedral", "key": "definition", "label": "Definition", "value": "The inverse image of the tetrahedral rotation group in SU(2); abstractly SL(2,3).", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups; binary tetrahedral example" }, { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#order", "subject": "group:binary-tetrahedral", "key": "order", "label": "Order", "value": "24", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#center", "subject": "group:binary-tetrahedral", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#abelian", "subject": "group:binary-tetrahedral", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#simple", "subject": "group:binary-tetrahedral", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#irreps", "subject": "group:binary-tetrahedral", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 2, 2, 2, 3", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#smallgroups-id", "subject": "group:binary-tetrahedral", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(24,3)", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#representations-1", "subject": "group:binary-tetrahedral", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#representations-2", "subject": "group:binary-tetrahedral", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#structure-1", "subject": "group:binary-tetrahedral", "key": "structure-1", "label": "Structure and conventions", "value": "The center acts trivially on the three singlets and the triplet; those representations descend to A₄.", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#relation-1", "subject": "group:binary-tetrahedral", "key": "relation-1", "label": "Family", "value": "family:binary-polyhedral", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#relation-2", "subject": "group:binary-tetrahedral", "key": "relation-2", "label": "2T / C₂ ≅ A₄", "value": "group:a-4", "evidence": [ { "source": "finite-groupnames-binary-tetrahedral", "locator": "Centre: G/Z" } ], "assumptions": [], "relationship": "central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-tetrahedral#relation-3", "subject": "group:binary-tetrahedral", "key": "relation-3", "label": "Binary tetrahedral embedding", "value": "group:su-2", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" } ], "assumptions": [], "relationship": "subgroup-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#definition", "subject": "group:binary-octahedral", "key": "definition", "label": "Definition", "value": "The inverse image of the octahedral rotation group in SU(2).", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" }, { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#order", "subject": "group:binary-octahedral", "key": "order", "label": "Order", "value": "48", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#center", "subject": "group:binary-octahedral", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#abelian", "subject": "group:binary-octahedral", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#simple", "subject": "group:binary-octahedral", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#irreps", "subject": "group:binary-octahedral", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 2, 2, 2, 3, 3, 4", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#smallgroups-id", "subject": "group:binary-octahedral", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(48,28)", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#representations-1", "subject": "group:binary-octahedral", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#representations-2", "subject": "group:binary-octahedral", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#structure-1", "subject": "group:binary-octahedral", "key": "structure-1", "label": "Structure and conventions", "value": "The faithful doublets and the four-dimensional irrep are quaternionic. The other irreps descend to the rotational quotient S₄.", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#relation-1", "subject": "group:binary-octahedral", "key": "relation-1", "label": "Family", "value": "family:binary-polyhedral", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#relation-2", "subject": "group:binary-octahedral", "key": "relation-2", "label": "2O / C₂ ≅ S₄", "value": "group:s-4", "evidence": [ { "source": "finite-groupnames-binary-octahedral", "locator": "Centre: G/Z" } ], "assumptions": [], "relationship": "central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-octahedral#relation-3", "subject": "group:binary-octahedral", "key": "relation-3", "label": "Binary octahedral embedding", "value": "group:su-2", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" } ], "assumptions": [], "relationship": "subgroup-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#definition", "subject": "group:binary-icosahedral", "key": "definition", "label": "Definition", "value": "The inverse image of the icosahedral rotation group in SU(2); abstractly SL(2,5).", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" }, { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#order", "subject": "group:binary-icosahedral", "key": "order", "label": "Order", "value": "120", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#center", "subject": "group:binary-icosahedral", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#abelian", "subject": "group:binary-icosahedral", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#simple", "subject": "group:binary-icosahedral", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#irreps", "subject": "group:binary-icosahedral", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 2, 2, 3, 3, 4, 4, 5, 6", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#smallgroups-id", "subject": "group:binary-icosahedral", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(120,5)", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#representations-1", "subject": "group:binary-icosahedral", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#representations-2", "subject": "group:binary-icosahedral", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#structure-1", "subject": "group:binary-icosahedral", "key": "structure-1", "label": "Structure and conventions", "value": "The two four-dimensional irreps differ: one is real and descends to A₅; the other is quaternionic and faithful.", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#relation-1", "subject": "group:binary-icosahedral", "key": "relation-1", "label": "Family", "value": "family:binary-polyhedral", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#relation-2", "subject": "group:binary-icosahedral", "key": "relation-2", "label": "2I / C₂ ≅ A₅", "value": "group:a-5", "evidence": [ { "source": "finite-groupnames-binary-icosahedral", "locator": "Centre: G/Z" } ], "assumptions": [], "relationship": "central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:binary-icosahedral#relation-3", "subject": "group:binary-icosahedral", "key": "relation-3", "label": "Binary icosahedral embedding", "value": "group:su-2", "evidence": [ { "source": "finite-taylor-mckay", "locator": "Binary polyhedral groups" } ], "assumptions": [], "relationship": "subgroup-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#definition", "subject": "group:psl-2-7", "key": "definition", "label": "Definition", "value": "SL(2,F₇)/{±I}; equivalently GL(3,F₂).", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Heading; Aliases; Generators and relations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#order", "subject": "group:psl-2-7", "key": "order", "label": "Order", "value": "168", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#center", "subject": "group:psl-2-7", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#abelian", "subject": "group:psl-2-7", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#simple", "subject": "group:psl-2-7", "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Simple label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#irreps", "subject": "group:psl-2-7", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 3, 3, 6, 7, 8", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#smallgroups-id", "subject": "group:psl-2-7", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(168,42)", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#representations-1", "subject": "group:psl-2-7", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#representations-2", "subject": "group:psl-2-7", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#structure-1", "subject": "group:psl-2-7", "key": "structure-1", "label": "Structure and conventions", "value": "The two triplets are inequivalent complex conjugates. The 6-, 7- and 8-dimensional irreps are real.", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-7#relation-1", "subject": "group:psl-2-7", "key": "relation-1", "label": "Family", "value": "family:finite-linear", "evidence": [ { "source": "finite-groupnames-psl-2-7", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#definition", "subject": "group:delta-27", "key": "definition", "label": "Definition", "value": "The order-27 subgroup of SU(3) generated by diag(ω,1,ω⁻¹), diag(1,ω,ω⁻¹) and the cyclic coordinate permutation, with ω³ = 1 and ω ≠ 1.", "evidence": [ { "source": "finite-benvenuti-delta27", "locator": "Equation (3.12)" }, { "source": "finite-groupnames-delta-27", "locator": "Heading and Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#order", "subject": "group:delta-27", "key": "order", "label": "Order", "value": "27", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#center", "subject": "group:delta-27", "key": "center", "label": "Center", "value": "C₃", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#abelian", "subject": "group:delta-27", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#simple", "subject": "group:delta-27", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Chief series; Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#irreps", "subject": "group:delta-27", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#smallgroups-id", "subject": "group:delta-27", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(27,3)", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#representations-1", "subject": "group:delta-27", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#representations-2", "subject": "group:delta-27", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#structure-1", "subject": "group:delta-27", "key": "structure-1", "label": "Structure and conventions", "value": "The center and commutator subgroup are both C₃; the abelianization is C₃ × C₃.", "evidence": [ { "source": "finite-groupnames-delta-27", "locator": "Centre; Commutator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#structure-2", "subject": "group:delta-27", "key": "structure-2", "label": "Structure and conventions", "value": "The SU(3) matrix realization is used in the cited study of invariant rings and orbifold gauge theories.", "evidence": [ { "source": "finite-benvenuti-delta27", "locator": "Section 3, equation (3.12) and following discussion" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:delta-27#relation-1", "subject": "group:delta-27", "key": "relation-1", "label": "Specified SU(3) embedding", "value": "group:su-3", "evidence": [ { "source": "finite-benvenuti-delta27", "locator": "Equation (3.12)" } ], "assumptions": [], "relationship": "subgroup-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#definition", "subject": "group:a-6", "key": "definition", "label": "Definition", "value": "The group of even permutations of six letters; also PSL(2,9).", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading and Aliases" }, { "source": "finite-craven-identifications", "locator": "Section 7, seven exceptional isomorphisms before Proposition 24" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#order", "subject": "group:a-6", "key": "order", "label": "Order", "value": "360", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading: order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#center", "subject": "group:a-6", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Centre" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#abelian", "subject": "group:a-6", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Commutator; character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#simple", "subject": "group:a-6", "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Simple label" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#irreps", "subject": "group:a-6", "key": "irreps", "label": "Ordinary complex irreducibles", "value": "1, 5, 5, 8, 8, 9, 10", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [ "Characteristic zero", "Finite-dimensional irreducibles over C", "Dimensions listed with multiplicity" ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#smallgroups-id", "subject": "group:a-6", "key": "smallgroups-id", "label": "GAP SmallGroups ID", "value": "SmallGroup(360,118)", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Aliases; GAP/Magma code by ID" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#representations-1", "subject": "group:a-6", "key": "representations-1", "label": "Representation scope", "value": "The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Character table" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#representations-2", "subject": "group:a-6", "key": "representations-2", "label": "Representation scope", "value": "Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Character table: identity column and real/complex/symplectic annotations" }, { "source": "gap", "locator": "Irr and section 71.12-5 Indicator" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#structure-1", "subject": "group:a-6", "key": "structure-1", "label": "Structure and conventions", "value": "Both five-dimensional irreps and both eight-dimensional irreps are distinct; dimension alone does not label an irrep.", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading; Centre; group-property labels; Aliases" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#relation-1", "subject": "group:a-6", "key": "relation-1", "label": "Family", "value": "family:finite-alternating", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:a-6#relation-2", "subject": "group:a-6", "key": "relation-2", "label": "Family", "value": "family:finite-linear", "evidence": [ { "source": "finite-groupnames-a-6", "locator": "Heading; Aliases" } ], "assumptions": [], "relationship": "member-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sporadic-m11#definition", "subject": "group:sporadic-m11", "key": "definition", "label": "Definition", "value": "The sporadic simple group M₁₁ identified by the ATLAS standard generators and group record.", "evidence": [ { "source": "finite-atlas-m11", "locator": "Heading and Order" }, { "source": "finite-atlas-m11", "locator": "Standard generators" }, { "source": "finite-atlas-sporadic-index", "locator": "M₁₁ entry" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sporadic-m11#order", "subject": "group:sporadic-m11", "key": "order", "label": "Order", "value": "7920", "evidence": [ { "source": "finite-atlas-m11", "locator": "Order" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sporadic-m11#simple", "subject": "group:sporadic-m11", "key": "simple", "label": "Simple", "value": "Yes", "evidence": [ { "source": "finite-atlas-sporadic-index", "locator": "M₁₁ entry in sporadic simple groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sporadic-m11#abelian", "subject": "group:sporadic-m11", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "finite-wilson-introduction", "locator": "CFSG statement, first slide" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sporadic-m11#representation-scope", "subject": "group:sporadic-m11", "key": "representation-scope", "label": "Representation coverage", "value": "Classification record; follow the ATLAS for available representations.", "evidence": [ { "source": "finite-atlas-m11", "locator": "Representations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sporadic-m11#scope-1", "subject": "group:sporadic-m11", "key": "scope-1", "label": "Reference scope", "value": "This entry establishes the group identity and exact order. 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Representation links upstream may refer to different fields, characteristics or central extensions; those choices must be checked before reuse.", "evidence": [ { "source": "finite-atlas-fi24-prime", "locator": "Representations and representation headings" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sporadic-fi24-prime#scope-2", "subject": "group:sporadic-fi24-prime", "key": "scope-2", "label": "Reference scope", "value": "The prime in Fi₂₄′ is part of the simple-group name. 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This is a simple real algebra, isomorphic to so(1,3); its complexification is sl(2,C) ⊕ sl(2,C).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: complex simple Lie algebras viewed as simple real algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#field", "subject": "algebra:sl-2-c-as-real", "key": "field", "label": "Scalar field", "value": "R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#dimension", "subject": "algebra:sl-2-c-as-real", "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#rank", "subject": "algebra:sl-2-c-as-real", "key": "rank", "label": "Absolute rank (of complexification)", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#simple", "subject": "algebra:sl-2-c-as-real", "key": "simple", "label": "Simple over R", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#compact", "subject": "algebra:sl-2-c-as-real", "key": "compact", "label": "Compact real algebra", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#complexification", "subject": "algebra:sl-2-c-as-real", "key": "complexification", "label": "Complexification", "value": "sl(2, C) ⊕ sl(2, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#real-rank", "subject": "algebra:sl-2-c-as-real", "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#integration-1", "subject": "algebra:sl-2-c-as-real", "key": "integration-1", "label": "From algebra to group", "value": "The linked groups integrate this real Lie algebra. They can have different centers and fundamental groups, so their ordinary representations are not interchangeable solely because their Lie algebras agree.", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:sl-2-c-as-real#integration-2", "subject": "algebra:sl-2-c-as-real", "key": "integration-2", "label": "From algebra to group", "value": "Restriction of scalars doubles the real dimension. Complexifying this realification gives two complex sl(2,C) factors, which label the two chiral highest weights. It is not a real form of the single complex A₁ algebra.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 9.4: real forms and complexification" }, { "source": "knapp-structure", "locator": "Theorem 6.105: complex simple algebras considered as real algebras" } ], "derivation": { "method": "For a complex algebra g, g_R ⊗_R C ≅ g ⊕ conjugate(g). 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The linked group records fix particular connected integrations, and state their global topology separately. 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It is isomorphic to so(2,4).", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#dimension", "subject": "real-form:su-2-2-r", "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#complexification", "subject": "real-form:su-2-2-r", "key": "complexification", "label": "Complexification", "value": "sl(4, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#compact", "subject": "real-form:su-2-2-r", "key": "compact", "label": "Compact real form", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#real-rank", "subject": "real-form:su-2-2-r", "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#global-forms-1", "subject": "real-form:su-2-2-r", "key": "global-forms-1", "label": "One algebra, several integrations", "value": "A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#relation-1", "subject": "real-form:su-2-2-r", "key": "relation-1", "label": "Actual real Lie algebra", "value": "algebra:su-2-2-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "realizes-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#relation-2", "subject": "real-form:su-2-2-r", "key": "relation-2", "label": "Complexification", "value": "algebra:a-3-c", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "real-form-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#relation-3", "subject": "real-form:su-2-2-r", "key": "relation-3", "label": "Curated global form", "value": "group:su-2-2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:su-2-2-r#relation-4", "subject": "real-form:su-2-2-r", "key": "relation-4", "label": "Curated global form", "value": "group:so-2-4-identity", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#definition", "subject": "real-form:sp-4-r", "key": "definition", "label": "Definition", "value": "The real 4 × 4 matrices X satisfying XᵀJ + JX = 0 for a nondegenerate alternating J. This split B₂ = C₂ algebra is isomorphic to so(2,3).", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#dimension", "subject": "real-form:sp-4-r", "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#complexification", "subject": "real-form:sp-4-r", "key": "complexification", "label": "Complexification", "value": "so(5, C) ≅ sp(4, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#compact", "subject": "real-form:sp-4-r", "key": "compact", "label": "Compact real form", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#real-rank", "subject": "real-form:sp-4-r", "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#global-forms-1", "subject": "real-form:sp-4-r", "key": "global-forms-1", "label": "One algebra, several integrations", "value": "A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#relation-1", "subject": "real-form:sp-4-r", "key": "relation-1", "label": "Actual real Lie algebra", "value": "algebra:sp-4-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "realizes-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#relation-2", "subject": "real-form:sp-4-r", "key": "relation-2", "label": "Complexification", "value": "algebra:b-2-c", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "real-form-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#relation-3", "subject": "real-form:sp-4-r", "key": "relation-3", "label": "Curated global form", "value": "group:sp-4-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sp-4-r#relation-4", "subject": "real-form:sp-4-r", "key": "relation-4", "label": "Curated global form", "value": "group:so-2-3-identity", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#definition", "subject": "real-form:sl-2-c-as-real", "key": "definition", "label": "Definition", "value": "The complex traceless 2 × 2 matrices with only real scalar multiplication retained. This is a simple real algebra, isomorphic to so(1,3); its complexification is sl(2,C) ⊕ sl(2,C).", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Theorem 6.105, printed page 362: complex simple Lie algebras viewed over R" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#dimension", "subject": "real-form:sl-2-c-as-real", "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#complexification", "subject": "real-form:sl-2-c-as-real", "key": "complexification", "label": "Complexification", "value": "sl(2, C) ⊕ sl(2, C)", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#compact", "subject": "real-form:sl-2-c-as-real", "key": "compact", "label": "Compact real form", "value": "No", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#real-rank", "subject": "real-form:sl-2-c-as-real", "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#global-forms-1", "subject": "real-form:sl-2-c-as-real", "key": "global-forms-1", "label": "One algebra, several integrations", "value": "A real form is algebraic data over R. The linked group records fix particular connected integrations, and state their global topology separately. Compactness of this real algebra should not be used to infer a particular quotient kernel.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#global-forms-2", "subject": "real-form:sl-2-c-as-real", "key": "global-forms-2", "label": "One algebra, several integrations", "value": "Here the complex algebra is A₁ ⊕ A₁. The same complexification also has the compact real form su(2) ⊕ su(2), but the two real algebras are different: the Lorentz algebra is simple over R and the compact one is a direct sum.", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "knapp-structure", "locator": "Theorem 6.105: real simple classification" } ], "derivation": { "method": "Compare the ideals of the realification with those of the compact direct sum, and complexify both.", "inputs": [ "algebra:sl-2-c-as-real", "algebra:so-4-compact-r", "algebra:a-1-plus-a-1-c" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#relation-1", "subject": "real-form:sl-2-c-as-real", "key": "relation-1", "label": "Actual real Lie algebra", "value": "algebra:sl-2-c-as-real", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "realizes-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#relation-2", "subject": "real-form:sl-2-c-as-real", "key": "relation-2", "label": "Complexification", "value": "algebra:a-1-plus-a-1-c", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "real-form-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#relation-3", "subject": "real-form:sl-2-c-as-real", "key": "relation-3", "label": "Curated global form", "value": "group:sl-2-c", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "real-form:sl-2-c-as-real#relation-4", "subject": "real-form:sl-2-c-as-real", "key": "relation-4", "label": "Curated global form", "value": "group:so-1-3-identity", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#definition", "subject": "group:su-4", "key": "definition", "label": "Definition", "value": "Complex unitary 4 × 4 matrices with determinant 1.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#dimension", "subject": "group:su-4", "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#rank", "subject": "group:su-4", "key": "rank", "label": "Compact rank", "value": "3", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#center", "subject": "group:su-4", "key": "center", "label": "Center", "value": "C₄", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#pi1", "subject": "group:su-4", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#compact", "subject": "group:su-4", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#connected", "subject": "group:su-4", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#abelian", "subject": "group:su-4", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#simple-algebra", "subject": "group:su-4", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#global-form", "subject": "group:su-4", "key": "global-form", "label": "Global form", "value": "Simply connected compact form of A₃", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#lie-algebra", "subject": "group:su-4", "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(4) ≅ so(6)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#topology-1", "subject": "group:su-4", "key": "topology-1", "label": "Global structure and topology", "value": "SU(4) is simply connected. Its center consists of the scalar fourth roots of unity. The identification Spin(6) ≅ SU(4) gives the vector cover SU(4) → SO(6) with kernel {±I}.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#representations-1", "subject": "group:su-4", "key": "representations-1", "label": "Representation category and descent", "value": "Finite-dimensional complex irreducibles have A₃ dominant integral highest weights. The defining 4 and its dual are the two chiral spin representations in the Spin(6) identification; the exterior square of the 4 is the vector 6.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "garrett-isogenies", "locator": "Section 2.9: SU(4) acting on the real form of its exterior square" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#relation-1", "subject": "group:su-4", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:su-4-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#relation-2", "subject": "group:su-4", "key": "relation-2", "label": "Real-form context", "value": "real-form:su-4-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-4#relation-3", "subject": "group:su-4", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:su-4-to-so-6", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad-centers", "locator": "Page 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphisms" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; 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Its adjoint global form is the quotient by this center.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-5#representations-1", "subject": "group:su-5", "key": "representations-1", "label": "Representation category and descent", "value": "Dominant integral A₄ highest weights label its irreducibles. 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The parity condition makes (−I,−1) act trivially. Its complex dimension is n + 1.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "derivation": { "method": "The diagonal kernel acts by (−1)ⁿ(−1)ᑫ, so descent requires n+q even.", "inputs": [ "SU(2) highest weight n has dimension n+1 and central sign (−1)ⁿ", "U(1) character z ↦ zᑫ", "U(2) quotient kernel {(I,1),(−I,−1)}" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:u-2#relation-1", "subject": "group:u-2", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:u-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-4#definition", "subject": "group:so-4", "key": "definition", "label": "Definition", "value": "The real orthogonal 4 × 4 matrices of determinant 1.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-4#dimension", "subject": "group:so-4", "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-4#rank", "subject": "group:so-4", "key": "rank", "label": "Compact rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-4#center", "subject": "group:so-4", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; 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Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#pi1", "subject": "group:spin-8", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#compact", "subject": "group:spin-8", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#connected", "subject": "group:spin-8", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#abelian", "subject": "group:spin-8", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#simple-algebra", "subject": "group:spin-8", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#global-form", "subject": "group:spin-8", "key": "global-form", "label": "Global form", "value": "Simply connected compact form of D₄", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; 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Its full center is C₂ × C₂, and quotienting by that full center gives PSO(8).", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#representations-1", "subject": "group:spin-8", "key": "representations-1", "label": "Representation category and descent", "value": "The vector representation and the two half-spin representations each have dimension 8. Triality permutes these representations and their order-two central kernels. Therefore the three intermediate central quotients are isomorphic as abstract groups, while their specified covering maps carry different representation information.", "evidence": [ { "source": "conrad", "locator": "Appendix P: spin representations; Remark T.2.1, printed page 262: D₄ triality and central kernels" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#relation-1", "subject": "group:spin-8", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:so-8-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#relation-2", "subject": "group:spin-8", "key": "relation-2", "label": "Real-form context", "value": "real-form:so-8-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-8#relation-3", "subject": "group:spin-8", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:spin-8-to-so-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; 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The remaining center {±I} gives the quotient PSO(8). Triality identifies other order-two Spin(8) quotients abstractly, without identifying their chosen vector or spinor actions.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-8#representations-1", "subject": "group:so-8", "key": "representations-1", "label": "Representation category and descent", "value": "Use finite-dimensional continuous complex representations. An irreducible representation of the simply connected compact cover descends precisely when the specified covering kernel acts trivially; the same Lie algebra does not remove that global condition.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-8#relation-1", "subject": "group:so-8", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:so-8-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-8#relation-2", "subject": "group:so-8", "key": "relation-2", "label": "Real-form context", "value": "real-form:so-8-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-8#relation-3", "subject": "group:so-8", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:spin-8-to-so-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-8#relation-4", "subject": "group:so-8", "key": "relation-4", "label": "Specified connected covering group", "value": "group:spin-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#definition", "subject": "group:pso-8", "key": "definition", "label": "Definition", "value": "The quotient of SO(8) by its center {±I}, equivalently the adjoint quotient of Spin(8).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#dimension", "subject": "group:pso-8", "key": "dimension", "label": "Real dimension", "value": "28", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#rank", "subject": "group:pso-8", "key": "rank", "label": "Compact rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#center", "subject": "group:pso-8", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#pi1", "subject": "group:pso-8", "key": "pi1", "label": "Fundamental group π₁", "value": "C₂ × C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#compact", "subject": "group:pso-8", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#connected", "subject": "group:pso-8", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#abelian", "subject": "group:pso-8", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#simple-algebra", "subject": "group:pso-8", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#global-form", "subject": "group:pso-8", "key": "global-form", "label": "Global form", "value": "Spin(8)/(C₂ × C₂)", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#lie-algebra", "subject": "group:pso-8", "key": "lie-algebra", "label": "Real Lie algebra", "value": "so(8)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#topology-1", "subject": "group:pso-8", "key": "topology-1", "label": "Global structure and topology", "value": "The simply connected cover is Spin(8), and the covering kernel is its full center C₂ × C₂. Thus the fundamental group has four elements and is not cyclic.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#representations-1", "subject": "group:pso-8", "key": "representations-1", "label": "Representation category and descent", "value": "A Spin(8) irrep descends to PSO(8) precisely when its highest weight lies in the D₄ root lattice. In particular, the vector and the two half-spin 8s do not descend, whereas the adjoint 28 does.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "conrad", "locator": "Appendix T.2, printed pages 261–262, especially Remark T.2.1: D₄ center, central kernels, and triality" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#relation-1", "subject": "group:pso-8", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:so-8-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#relation-2", "subject": "group:pso-8", "key": "relation-2", "label": "Real-form context", "value": "real-form:so-8-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:pso-8#relation-3", "subject": "group:pso-8", "key": "relation-3", "label": "Specified connected covering group", "value": "group:spin-8", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#definition", "subject": "group:spin-10", "key": "definition", "label": "Definition", "value": "The compact spin group in the even Clifford algebra of Euclidean R¹⁰, with its standard vector action.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#dimension", "subject": "group:spin-10", "key": "dimension", "label": "Real dimension", "value": "45", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#rank", "subject": "group:spin-10", "key": "rank", "label": "Compact rank", "value": "5", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#center", "subject": "group:spin-10", "key": "center", "label": "Center", "value": "C₄", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#pi1", "subject": "group:spin-10", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#compact", "subject": "group:spin-10", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#connected", "subject": "group:spin-10", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#abelian", "subject": "group:spin-10", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#simple-algebra", "subject": "group:spin-10", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#global-form", "subject": "group:spin-10", "key": "global-form", "label": "Global form", "value": "Simply connected compact form of D₅", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#lie-algebra", "subject": "group:spin-10", "key": "lie-algebra", "label": "Real Lie algebra", "value": "so(10)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#topology-1", "subject": "group:spin-10", "key": "topology-1", "label": "Global structure and topology", "value": "Spin(10) is simply connected and its center is cyclic of order 4. The element −1 is the order-two kernel of the vector cover to SO(10); quotienting by it leaves an order-two center.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#representations-1", "subject": "group:spin-10", "key": "representations-1", "label": "Representation category and descent", "value": "The vector representation has dimension 10 and factors through SO(10). The two complex half-spin representations have dimension 16 and do not factor through SO(10), because the covering element −1 acts as minus the identity on spinors.", "evidence": [ { "source": "conrad", "locator": "Appendix P: Clifford modules and half-spin representations; dim S± = 2^(r−1) in dimension 2r" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#relation-1", "subject": "group:spin-10", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:so-10-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#relation-2", "subject": "group:spin-10", "key": "relation-2", "label": "Real-form context", "value": "real-form:so-10-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:spin-10#relation-3", "subject": "group:spin-10", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:spin-10-to-so-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#definition", "subject": "group:so-10", "key": "definition", "label": "Definition", "value": "The real orthogonal 10 × 10 matrices of determinant 1.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#dimension", "subject": "group:so-10", "key": "dimension", "label": "Real dimension", "value": "45", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#rank", "subject": "group:so-10", "key": "rank", "label": "Compact rank", "value": "5", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#center", "subject": "group:so-10", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#pi1", "subject": "group:so-10", "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#compact", "subject": "group:so-10", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#connected", "subject": "group:so-10", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#abelian", "subject": "group:so-10", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#simple-algebra", "subject": "group:so-10", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#global-form", "subject": "group:so-10", "key": "global-form", "label": "Global form", "value": "Spin(10)/{±1}", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#lie-algebra", "subject": "group:so-10", "key": "lie-algebra", "label": "Real Lie algebra", "value": "so(10)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#topology-1", "subject": "group:so-10", "key": "topology-1", "label": "Global structure and topology", "value": "SO(10) has universal cover Spin(10) and fundamental group C₂. Its matrix center is {±I}. A statement about a spinor 16 must therefore specify the Spin(10) cover.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#representations-1", "subject": "group:so-10", "key": "representations-1", "label": "Representation category and descent", "value": "Use finite-dimensional continuous complex representations. An irreducible representation of the simply connected compact cover descends precisely when the specified covering kernel acts trivially; the same Lie algebra does not remove that global condition.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#relation-1", "subject": "group:so-10", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:so-10-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#relation-2", "subject": "group:so-10", "key": "relation-2", "label": "Real-form context", "value": "real-form:so-10-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#relation-3", "subject": "group:so-10", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:spin-10-to-so-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-10#relation-4", "subject": "group:so-10", "key": "relation-4", "label": "Specified connected covering group", "value": "group:spin-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:g-2-compact#definition", "subject": "group:g-2-compact", "key": "definition", "label": "Definition", "value": "The automorphism group of the real octonion division algebra O.", "evidence": [ { "source": "baez-octonions", "locator": "Section 4.1: G₂ as octonion automorphisms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:g-2-compact#dimension", "subject": "group:g-2-compact", "key": "dimension", "label": "Real dimension", "value": "14", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:g-2-compact#rank", "subject": "group:g-2-compact", "key": "rank", "label": "Compact rank", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:g-2-compact#center", "subject": "group:g-2-compact", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; 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Their action on Im(O) is a faithful real representation of dimension 7, preserving its Euclidean norm and octonion multiplication.", "evidence": [ { "source": "baez-octonions", "locator": "Section 4.1: action on imaginary octonions and the subgroup G₂ ⊂ SO(7)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:g-2-compact#relation-1", "subject": "group:g-2-compact", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:g-2-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:g-2-compact#relation-2", "subject": "group:g-2-compact", "key": "relation-2", "label": "Real-form context", "value": "real-form:g-2-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:f-4-compact#definition", "subject": "group:f-4-compact", "key": "definition", "label": "Definition", "value": "The automorphism group of the Euclidean exceptional Jordan algebra of Hermitian 3 × 3 octonionic matrices, with product X ∘ Y = (XY + YX)/2.", "evidence": [ { "source": "baez-octonions", "locator": "Section 4.2: F₄ and the exceptional Jordan algebra" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:f-4-compact#dimension", "subject": "group:f-4-compact", "key": "dimension", "label": "Real dimension", "value": "52", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:f-4-compact#rank", "subject": "group:f-4-compact", "key": "rank", "label": "Compact rank", "value": "4", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:f-4-compact#center", "subject": "group:f-4-compact", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; 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Its compact global form is therefore unique for this Lie algebra.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:f-4-compact#representations-1", "subject": "group:f-4-compact", "key": "representations-1", "label": "Representation category and descent", "value": "The action on the exceptional Jordan algebra fixes its identity. 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Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#pi1", "subject": "group:e-6-compact-sc", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#compact", "subject": "group:e-6-compact-sc", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#connected", "subject": "group:e-6-compact-sc", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#abelian", "subject": "group:e-6-compact-sc", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#simple-algebra", "subject": "group:e-6-compact-sc", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#global-form", "subject": "group:e-6-compact-sc", "key": "global-form", "label": "Global form", "value": "Simply connected compact form of E₆", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#lie-algebra", "subject": "group:e-6-compact-sc", "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₆ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#topology-1", "subject": "group:e-6-compact-sc", "key": "topology-1", "label": "Global structure and topology", "value": "The center is C₃. Its only nontrivial connected central quotient is the adjoint compact E₆/C₃, whose fundamental group is C₃.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#representations-1", "subject": "group:e-6-compact-sc", "key": "representations-1", "label": "Representation category and descent", "value": "Dominant integral E₆ highest weights classify the finite-dimensional complex irreducibles. Passing to the adjoint form retains precisely the weights in the root lattice.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#relation-1", "subject": "group:e-6-compact-sc", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:e-6-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-sc#relation-2", "subject": "group:e-6-compact-sc", "key": "relation-2", "label": "Real-form context", "value": "real-form:e-6-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#definition", "subject": "group:e-7-compact-sc", "key": "definition", "label": "Definition", "value": "The connected simply connected compact Lie group integrating the compact real algebra of type E₇.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#dimension", "subject": "group:e-7-compact-sc", "key": "dimension", "label": "Real dimension", "value": "133", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#rank", "subject": "group:e-7-compact-sc", "key": "rank", "label": "Compact rank", "value": "7", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#center", "subject": "group:e-7-compact-sc", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#pi1", "subject": "group:e-7-compact-sc", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#compact", "subject": "group:e-7-compact-sc", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#connected", "subject": "group:e-7-compact-sc", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#abelian", "subject": "group:e-7-compact-sc", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#simple-algebra", "subject": "group:e-7-compact-sc", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#global-form", "subject": "group:e-7-compact-sc", "key": "global-form", "label": "Global form", "value": "Simply connected compact form of E₇", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#lie-algebra", "subject": "group:e-7-compact-sc", "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₇ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#topology-1", "subject": "group:e-7-compact-sc", "key": "topology-1", "label": "Global structure and topology", "value": "The center is C₂. Quotienting by it gives the adjoint compact E₇/C₂, with fundamental group C₂.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#representations-1", "subject": "group:e-7-compact-sc", "key": "representations-1", "label": "Representation category and descent", "value": "Dominant integral E₇ highest weights classify finite-dimensional complex irreducibles. An irrep descends to the adjoint group exactly when its central sign is positive, equivalently when its highest weight lies in the root lattice.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#relation-1", "subject": "group:e-7-compact-sc", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:e-7-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-sc#relation-2", "subject": "group:e-7-compact-sc", "key": "relation-2", "label": "Real-form context", "value": "real-form:e-7-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#definition", "subject": "group:e-8-compact", "key": "definition", "label": "Definition", "value": "The connected compact Lie group with compact real Lie algebra of type E₈; its simply connected and adjoint forms agree.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#dimension", "subject": "group:e-8-compact", "key": "dimension", "label": "Real dimension", "value": "248", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#rank", "subject": "group:e-8-compact", "key": "rank", "label": "Compact rank", "value": "8", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#center", "subject": "group:e-8-compact", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#pi1", "subject": "group:e-8-compact", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#compact", "subject": "group:e-8-compact", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#connected", "subject": "group:e-8-compact", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#abelian", "subject": "group:e-8-compact", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#simple-algebra", "subject": "group:e-8-compact", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#global-form", "subject": "group:e-8-compact", "key": "global-form", "label": "Global form", "value": "Simply connected and adjoint compact form of E₈", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#lie-algebra", "subject": "group:e-8-compact", "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₈ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#topology-1", "subject": "group:e-8-compact", "key": "topology-1", "label": "Global structure and topology", "value": "Its center and fundamental group are both trivial. There is no nontrivial connected central quotient of the simply connected compact E₈ group.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#representations-1", "subject": "group:e-8-compact", "key": "representations-1", "label": "Representation category and descent", "value": "The E₈ weight lattice equals its root lattice. Thus every dominant integral highest weight integrates to this single compact global form; its adjoint representation has dimension 248.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#relation-1", "subject": "group:e-8-compact", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:e-8-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-8-compact#relation-2", "subject": "group:e-8-compact", "key": "relation-2", "label": "Real-form context", "value": "real-form:e-8-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#definition", "subject": "group:e-6-compact-ad", "key": "definition", "label": "Definition", "value": "The quotient of the simply connected compact E₆ group by its full center C₃.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#dimension", "subject": "group:e-6-compact-ad", "key": "dimension", "label": "Real dimension", "value": "78", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#rank", "subject": "group:e-6-compact-ad", "key": "rank", "label": "Compact rank", "value": "6", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#center", "subject": "group:e-6-compact-ad", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#pi1", "subject": "group:e-6-compact-ad", "key": "pi1", "label": "Fundamental group π₁", "value": "C₃", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#compact", "subject": "group:e-6-compact-ad", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#connected", "subject": "group:e-6-compact-ad", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#abelian", "subject": "group:e-6-compact-ad", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#simple-algebra", "subject": "group:e-6-compact-ad", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#global-form", "subject": "group:e-6-compact-ad", "key": "global-form", "label": "Global form", "value": "E₆ simply connected / C₃", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#lie-algebra", "subject": "group:e-6-compact-ad", "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₆ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#topology-1", "subject": "group:e-6-compact-ad", "key": "topology-1", "label": "Global structure and topology", "value": "The adjoint group has trivial center and fundamental group C₃. The cover has the same local real algebra and dimension.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#representations-1", "subject": "group:e-6-compact-ad", "key": "representations-1", "label": "Representation category and descent", "value": "Finite-dimensional complex irreducibles are the dominant E₆ weights lying in the root lattice. This is the zero-center-character sector of the simply connected group.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#relation-1", "subject": "group:e-6-compact-ad", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:e-6-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#relation-2", "subject": "group:e-6-compact-ad", "key": "relation-2", "label": "Real-form context", "value": "real-form:e-6-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-6-compact-ad#relation-3", "subject": "group:e-6-compact-ad", "key": "relation-3", "label": "Specified connected covering group", "value": "group:e-6-compact-sc", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#definition", "subject": "group:e-7-compact-ad", "key": "definition", "label": "Definition", "value": "The quotient of the simply connected compact E₇ group by its full center C₂.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#dimension", "subject": "group:e-7-compact-ad", "key": "dimension", "label": "Real dimension", "value": "133", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#rank", "subject": "group:e-7-compact-ad", "key": "rank", "label": "Compact rank", "value": "7", "evidence": [ { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#center", "subject": "group:e-7-compact-ad", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#pi1", "subject": "group:e-7-compact-ad", "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#compact", "subject": "group:e-7-compact-ad", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#connected", "subject": "group:e-7-compact-ad", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lectures 26–28 and Appendix V: compact semisimple groups and highest weights" }, { "source": "conrad", "locator": "Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#abelian", "subject": "group:e-7-compact-ad", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#simple-algebra", "subject": "group:e-7-compact-ad", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#global-form", "subject": "group:e-7-compact-ad", "key": "global-form", "label": "Global form", "value": "E₇ simply connected / C₂", "evidence": [ { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#lie-algebra", "subject": "group:e-7-compact-ad", "key": "lie-algebra", "label": "Real Lie algebra", "value": "e₇ compact", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#topology-1", "subject": "group:e-7-compact-ad", "key": "topology-1", "label": "Global structure and topology", "value": "The adjoint group has trivial center and fundamental group C₂. This quotient changes global representation descent while preserving the local Lie algebra.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#representations-1", "subject": "group:e-7-compact-ad", "key": "representations-1", "label": "Representation category and descent", "value": "Finite-dimensional complex irreducibles are the dominant E₇ weights lying in the root lattice, equivalently those on which the nonidentity central element of the cover acts trivially.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#relation-1", "subject": "group:e-7-compact-ad", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:e-7-compact-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#relation-2", "subject": "group:e-7-compact-ad", "key": "relation-2", "label": "Real-form context", "value": "real-form:e-7-compact-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:e-7-compact-ad#relation-3", "subject": "group:e-7-compact-ad", "key": "relation-3", "label": "Specified connected covering group", "value": "group:e-7-compact-sc", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#definition", "subject": "group:sl-2-r", "key": "definition", "label": "Definition", "value": "The real 2 × 2 matrices of determinant 1.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#dimension", "subject": "group:sl-2-r", "key": "dimension", "label": "Real dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#real-rank", "subject": "group:sl-2-r", "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#center", "subject": "group:sl-2-r", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#pi1", "subject": "group:sl-2-r", "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(2) ≅ U(1)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#compact", "subject": "group:sl-2-r", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#connected", "subject": "group:sl-2-r", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#abelian", "subject": "group:sl-2-r", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#simple-algebra", "subject": "group:sl-2-r", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#global-form", "subject": "group:sl-2-r", "key": "global-form", "label": "Global form", "value": "Matrix form, a quotient of the universal cover by 2Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#lie-algebra", "subject": "group:sl-2-r", "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(2, R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#maximal-compact", "subject": "group:sl-2-r", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(2) ≅ U(1)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#topology-1", "subject": "group:sl-2-r", "key": "topology-1", "label": "Global structure and topology", "value": "Polar decomposition gives the manifold SO(2) × R², so the fundamental group is Z. The center is {±I}. The isomorphism with SU(1,1) is an isomorphism of real matrix Lie groups after a complex change of basis.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#representations-1", "subject": "group:sl-2-r", "key": "representations-1", "label": "Representation category and descent", "value": "Finite-dimensional complex irreducible representations are restrictions of sl(2,C) highest-weight modules with integer highest weight n ≥ 0 and dimension n + 1. The central element −I acts by (−1)ⁿ. These are distinct from infinite-dimensional unitary representations.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-lie-notes", "locator": "Remark 36.7, printed page 196: finite-dimensional representations of the universal cover factor through SL(2,R)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#relation-1", "subject": "group:sl-2-r", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#relation-2", "subject": "group:sl-2-r", "key": "relation-2", "label": "Real-form context", "value": "real-form:sl-2-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#relation-3", "subject": "group:sl-2-r", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:sl-2-r-to-psl-2-r", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#relation-4", "subject": "group:sl-2-r", "key": "relation-4", "label": "Explicit homomorphism", "value": "map:universal-sl-2-r-to-sl-2-r", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-2-r#relation-5", "subject": "group:sl-2-r", "key": "relation-5", "label": "Specified connected covering group", "value": "group:universal-cover-sl-2-r", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#definition", "subject": "group:psl-2-r", "key": "definition", "label": "Definition", "value": "The quotient of SL(2,R) by its scalar center {±I}.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#dimension", "subject": "group:psl-2-r", "key": "dimension", "label": "Real dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#real-rank", "subject": "group:psl-2-r", "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#center", "subject": "group:psl-2-r", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#pi1", "subject": "group:psl-2-r", "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(2), with the adjoint angular period", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#compact", "subject": "group:psl-2-r", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#connected", "subject": "group:psl-2-r", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#abelian", "subject": "group:psl-2-r", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#simple-algebra", "subject": "group:psl-2-r", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#global-form", "subject": "group:psl-2-r", "key": "global-form", "label": "Global form", "value": "SL(2,R)/{±I}", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.2: the SL(2,R) orthogonal action of signature (2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#lie-algebra", "subject": "group:psl-2-r", "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(2, R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#maximal-compact", "subject": "group:psl-2-r", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(2), with the adjoint angular period", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#topology-1", "subject": "group:psl-2-r", "key": "topology-1", "label": "Global structure and topology", "value": "Its maximal compact subgroup is again a circle, so it has fundamental group Z. The adjoint action identifies it with SO₀(2,1), the identity component; the notation does not include the other components of O(2,1).", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.2: the SL(2,R) orthogonal action of signature (2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#representations-1", "subject": "group:psl-2-r", "key": "representations-1", "label": "Representation category and descent", "value": "An SL(2,R) finite-dimensional complex highest-weight module descends precisely for even highest weight n, so its dimension n + 1 is odd. The complete unitary dual is a different reference problem.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "etingof-lie-notes", "locator": "Remark 36.7: finite-dimensional representations and the universal cover" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#relation-1", "subject": "group:psl-2-r", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#relation-2", "subject": "group:psl-2-r", "key": "relation-2", "label": "Real-form context", "value": "real-form:sl-2-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#relation-3", "subject": "group:psl-2-r", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:sl-2-r-to-psl-2-r", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.2: the SL(2,R) orthogonal action of signature (2,1)" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:psl-2-r#relation-4", "subject": "group:psl-2-r", "key": "relation-4", "label": "Specified connected covering group", "value": "group:universal-cover-sl-2-r", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.2: the SL(2,R) orthogonal action of signature (2,1)" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#definition", "subject": "group:universal-cover-sl-2-r", "key": "definition", "label": "Definition", "value": "The simply connected covering Lie group of SL(2,R), or equivalently of PSL(2,R).", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: the universal covering group of SL(2,R)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#dimension", "subject": "group:universal-cover-sl-2-r", "key": "dimension", "label": "Real dimension", "value": "3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#real-rank", "subject": "group:universal-cover-sl-2-r", "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#center", "subject": "group:universal-cover-sl-2-r", "key": "center", "label": "Center", "value": "Z", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: infinite cyclic center and absence of nontrivial compact subgroups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#pi1", "subject": "group:universal-cover-sl-2-r", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: universal covering Lie groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#compact", "subject": "group:universal-cover-sl-2-r", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#connected", "subject": "group:universal-cover-sl-2-r", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#abelian", "subject": "group:universal-cover-sl-2-r", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#simple-algebra", "subject": "group:universal-cover-sl-2-r", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#global-form", "subject": "group:universal-cover-sl-2-r", "key": "global-form", "label": "Global form", "value": "Simply connected real Lie group; not a matrix group", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: structure of the universal cover" }, { "source": "knapp-structure", "locator": "Theorem 6.46: Iwasawa decomposition" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#lie-algebra", "subject": "group:universal-cover-sl-2-r", "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(2, R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#maximal-compact", "subject": "group:universal-cover-sl-2-r", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "Trivial", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: infinite cyclic center and absence of nontrivial compact subgroups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#topology-1", "subject": "group:universal-cover-sl-2-r", "key": "topology-1", "label": "Global structure and topology", "value": "The lifted Iwasawa factors identify its manifold with R³. Let z be the central element represented by a lift of a rotation through π in SL(2,R). Then the projection to SL(2,R) has kernel ⟨z²⟩, while the projection to PSL(2,R) has kernel ⟨z⟩.", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: structure of the universal cover" }, { "source": "knapp-structure", "locator": "Theorem 6.46: Iwasawa decomposition" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#representations-1", "subject": "group:universal-cover-sl-2-r", "key": "representations-1", "label": "Representation category and descent", "value": "Every finite-dimensional complex representation factors through SL(2,R). Consequently this simply connected Lie group has no faithful finite-dimensional matrix representation; its infinite cyclic center is not detected faithfully in that category.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Remark 36.7, printed page 196; Exercise 11.20: finite-dimensional representations and nonlinearity of the universal cover" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#relation-1", "subject": "group:universal-cover-sl-2-r", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#relation-2", "subject": "group:universal-cover-sl-2-r", "key": "relation-2", "label": "Real-form context", "value": "real-form:sl-2-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:universal-cover-sl-2-r#relation-3", "subject": "group:universal-cover-sl-2-r", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:universal-sl-2-r-to-sl-2-r", "evidence": [ { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: structure of the universal cover" }, { "source": "knapp-structure", "locator": "Theorem 6.46: Iwasawa decomposition" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#definition", "subject": "group:sl-3-r", "key": "definition", "label": "Definition", "value": "The real 3 × 3 matrices of determinant 1.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#dimension", "subject": "group:sl-3-r", "key": "dimension", "label": "Real dimension", "value": "8", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#real-rank", "subject": "group:sl-3-r", "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#center", "subject": "group:sl-3-r", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#pi1", "subject": "group:sl-3-r", "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(3)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#compact", "subject": "group:sl-3-r", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#connected", "subject": "group:sl-3-r", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#abelian", "subject": "group:sl-3-r", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#simple-algebra", "subject": "group:sl-3-r", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#global-form", "subject": "group:sl-3-r", "key": "global-form", "label": "Global form", "value": "Split real matrix form of A₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#lie-algebra", "subject": "group:sl-3-r", "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(3, R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#maximal-compact", "subject": "group:sl-3-r", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#topology-1", "subject": "group:sl-3-r", "key": "topology-1", "label": "Global structure and topology", "value": "Polar decomposition identifies the manifold with SO(3) × R⁵. Therefore π₁ is C₂ and this centerless matrix group is not simply connected. Real scalar matrices of determinant 1 give only the identity in its center.", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#representations-1", "subject": "group:sl-3-r", "key": "representations-1", "label": "Representation category and descent", "value": "The defining real 3 gives a faithful matrix representation. Finite-dimensional complex irreducibles are restrictions of sl(3,C) highest-weight modules; representation data for infinite-dimensional unitary families are outside this record.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-lie-notes", "locator": "Section 9: Lie-group and Lie-algebra representations; connected integration" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#relation-1", "subject": "group:sl-3-r", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:sl-3-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sl-3-r#relation-2", "subject": "group:sl-3-r", "key": "relation-2", "label": "Real-form context", "value": "real-form:sl-3-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#definition", "subject": "group:su-2-2", "key": "definition", "label": "Definition", "value": "The complex 4 × 4 matrices A with det A = 1 and A†HA = H, where H = diag(1,1,−1,−1).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#dimension", "subject": "group:su-2-2", "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#real-rank", "subject": "group:su-2-2", "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#center", "subject": "group:su-2-2", "key": "center", "label": "Center", "value": "C₄", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#pi1", "subject": "group:su-2-2", "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: S(U(2) × U(2))", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#compact", "subject": "group:su-2-2", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#connected", "subject": "group:su-2-2", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#abelian", "subject": "group:su-2-2", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#simple-algebra", "subject": "group:su-2-2", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#global-form", "subject": "group:su-2-2", "key": "global-form", "label": "Global form", "value": "Connected spin double cover of SO₀(2,4); not its universal cover", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#lie-algebra", "subject": "group:su-2-2", "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(2,2) ≅ so(2,4)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#maximal-compact", "subject": "group:su-2-2", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "S(U(2) × U(2))", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#topology-1", "subject": "group:su-2-2", "key": "topology-1", "label": "Global structure and topology", "value": "The maximal compact subgroup is S(U(2) × U(2)), whose determinant winding gives fundamental group Z. The scalar center is C₄. The orthogonal action on a real form of Λ²C⁴ gives a double cover of SO₀(2,4) with kernel {±I}; SU(2,2) itself is not simply connected.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#representations-1", "subject": "group:su-2-2", "key": "representations-1", "label": "Representation category and descent", "value": "The defining complex 4 is a chiral spinor for the connected orthogonal cover. Its exterior square has an invariant real structure and gives the real vector 6 of signature (2,4). Because −I acts nontrivially on the 4, that 4 does not descend to SO₀(2,4).", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11: exterior-square construction and double-cover kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#relation-1", "subject": "group:su-2-2", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:su-2-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#relation-2", "subject": "group:su-2-2", "key": "relation-2", "label": "Real-form context", "value": "real-form:su-2-2-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:su-2-2#relation-3", "subject": "group:su-2-2", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:su-2-2-to-so-2-4", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#definition", "subject": "group:sp-4-r", "key": "definition", "label": "Definition", "value": "The real 4 × 4 matrices A satisfying AᵀJA = J for a fixed nondegenerate alternating matrix J.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#dimension", "subject": "group:sp-4-r", "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#real-rank", "subject": "group:sp-4-r", "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#center", "subject": "group:sp-4-r", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#pi1", "subject": "group:sp-4-r", "key": "pi1", "label": "Fundamental group π₁", "value": "Z", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: U(2)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#compact", "subject": "group:sp-4-r", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#connected", "subject": "group:sp-4-r", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#abelian", "subject": "group:sp-4-r", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#simple-algebra", "subject": "group:sp-4-r", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#global-form", "subject": "group:sp-4-r", "key": "global-form", "label": "Global form", "value": "Connected spin double cover of SO₀(2,3); not its universal cover", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#lie-algebra", "subject": "group:sp-4-r", "key": "lie-algebra", "label": "Real Lie algebra", "value": "sp(4, R) ≅ so(2,3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#maximal-compact", "subject": "group:sp-4-r", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "U(2)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#topology-1", "subject": "group:sp-4-r", "key": "topology-1", "label": "Global structure and topology", "value": "Cartan decomposition retracts Sp(4,R) onto U(2), giving fundamental group Z. Its center is {±I}. Its orthogonal five-dimensional action is a double cover of SO₀(2,3), whose fundamental group is Z × C₂. This real group is different from compact Sp(2).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#representations-1", "subject": "group:sp-4-r", "key": "representations-1", "label": "Representation category and descent", "value": "The real defining 4 preserves its symplectic form. The primitive part of Λ²R⁴ has dimension 5, carries an invariant quadratic form of signature (2,3) up to overall sign, and gives the double-cover map.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: five-dimensional orthogonal construction for the real rank-two symplectic group" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#notation-1", "subject": "group:sp-4-r", "key": "notation-1", "label": "Symplectic notation", "value": "This site writes compact Sp(n) for quaternionic n × n unitary matrices, also called USp(2n). It writes Sp(2n,R) for real matrices of size 2n preserving an alternating form. The two conventions count different dimensions.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#relation-1", "subject": "group:sp-4-r", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:sp-4-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#relation-2", "subject": "group:sp-4-r", "key": "relation-2", "label": "Real-form context", "value": "real-form:sp-4-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:sp-4-r#relation-3", "subject": "group:sp-4-r", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:sp-4-r-to-so-2-3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#definition", "subject": "group:so-1-3-identity", "key": "definition", "label": "Definition", "value": "The identity component of the real determinant-one matrices preserving a quadratic form with signature (1,3).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#dimension", "subject": "group:so-1-3-identity", "key": "dimension", "label": "Real dimension", "value": "6", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#real-rank", "subject": "group:so-1-3-identity", "key": "real-rank", "label": "Real rank", "value": "1", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#center", "subject": "group:so-1-3-identity", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#pi1", "subject": "group:so-1-3-identity", "key": "pi1", "label": "Fundamental group π₁", "value": "C₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(3)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#compact", "subject": "group:so-1-3-identity", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#connected", "subject": "group:so-1-3-identity", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#abelian", "subject": "group:so-1-3-identity", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#simple-algebra", "subject": "group:so-1-3-identity", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#global-form", "subject": "group:so-1-3-identity", "key": "global-form", "label": "Global form", "value": "SL(2,C)/{±I}", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) covering SO(3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#lie-algebra", "subject": "group:so-1-3-identity", "key": "lie-algebra", "label": "Real Lie algebra", "value": "sl(2, C) viewed over R", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#maximal-compact", "subject": "group:so-1-3-identity", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#topology-1", "subject": "group:so-1-3-identity", "key": "topology-1", "label": "Global structure and topology", "value": "Its maximal compact subgroup is SO(3), and its manifold is SO(3) × R³. The simply connected double cover is SL(2,C), acting on Hermitian 2 × 2 matrices by X ↦ AXA†.", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) covering SO(3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#representations-1", "subject": "group:so-1-3-identity", "key": "representations-1", "label": "Representation category and descent", "value": "Finite-dimensional complex irreducibles of the SL(2,C) cover have labels (jL,jR). They descend to SO₀(1,3) exactly when jL + jR is an integer. The two Weyl spinors therefore belong to the cover, while the vector (1/2,1/2) descends.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "knapp-structure", "locator": "Theorem 6.105: complex simple algebras viewed as real" } ], "derivation": { "method": "The complexified Lorentz algebra is sl(2,C) ⊕ sl(2,C). The covering element −I acts with product sign (−1)^(2jL+2jR).", "inputs": [ "algebra:sl-2-c-as-real", "SL(2,C) → SO₀(1,3) has kernel {±I}" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#relation-1", "subject": "group:so-1-3-identity", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:sl-2-c-as-real", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#relation-2", "subject": "group:so-1-3-identity", "key": "relation-2", "label": "Real-form context", "value": "real-form:sl-2-c-as-real", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#relation-3", "subject": "group:so-1-3-identity", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:sl-2-c-to-lorentz", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) covering SO(3,1)" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-1-3-identity#relation-4", "subject": "group:so-1-3-identity", "key": "relation-4", "label": "Specified connected covering group", "value": "group:sl-2-c", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) covering SO(3,1)" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#definition", "subject": "group:so-2-4-identity", "key": "definition", "label": "Definition", "value": "The identity component of the determinant-one real 6 × 6 matrices preserving a quadratic form with signature (2,4).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#dimension", "subject": "group:so-2-4-identity", "key": "dimension", "label": "Real dimension", "value": "15", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#real-rank", "subject": "group:so-2-4-identity", "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#center", "subject": "group:so-2-4-identity", "key": "center", "label": "Center", "value": "C₂", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#pi1", "subject": "group:so-2-4-identity", "key": "pi1", "label": "Fundamental group π₁", "value": "Z × C₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(2) × SO(4)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#compact", "subject": "group:so-2-4-identity", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#connected", "subject": "group:so-2-4-identity", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#abelian", "subject": "group:so-2-4-identity", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#simple-algebra", "subject": "group:so-2-4-identity", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#global-form", "subject": "group:so-2-4-identity", "key": "global-form", "label": "Global form", "value": "SU(2,2)/{±I}", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.11: SU(2,2) double cover" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#lie-algebra", "subject": "group:so-2-4-identity", "key": "lie-algebra", "label": "Real Lie algebra", "value": "su(2,2) ≅ so(2,4)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#maximal-compact", "subject": "group:so-2-4-identity", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(2) × SO(4)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#topology-1", "subject": "group:so-2-4-identity", "key": "topology-1", "label": "Global structure and topology", "value": "A Cartan decomposition gives maximal compact subgroup SO(2) × SO(4), hence π₁ = Z × C₂. The matrix −I lies in the identity component because both signature blocks have even dimension; the center is {±I}. Its spin double cover SU(2,2) still has π₁ = Z.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.11: SU(2,2) double cover" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#representations-1", "subject": "group:so-2-4-identity", "key": "representations-1", "label": "Representation category and descent", "value": "The vector 6 is an ordinary representation of this matrix group. The chiral spinor 4 of SU(2,2) does not descend through the order-two covering kernel. For a spacetime application, specify separately whether the action uses this group, its projective quotient, or a further covering group.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11: exterior-square orthogonal action and its kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#relation-1", "subject": "group:so-2-4-identity", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:su-2-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#relation-2", "subject": "group:so-2-4-identity", "key": "relation-2", "label": "Real-form context", "value": "real-form:su-2-2-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#relation-3", "subject": "group:so-2-4-identity", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:su-2-2-to-so-2-4", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.11: SU(2,2) double cover" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-4-identity#relation-4", "subject": "group:so-2-4-identity", "key": "relation-4", "label": "Specified connected covering group", "value": "group:su-2-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.11: SU(2,2) double cover" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#definition", "subject": "group:so-2-3-identity", "key": "definition", "label": "Definition", "value": "The identity component of the determinant-one real 5 × 5 matrices preserving a quadratic form with signature (2,3).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#dimension", "subject": "group:so-2-3-identity", "key": "dimension", "label": "Real dimension", "value": "10", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" }, { "source": "etingof-coxeter", "locator": "Section 32.4, printed pages 175–176: Coxeter and dual Coxeter numbers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#real-rank", "subject": "group:so-2-3-identity", "key": "real-rank", "label": "Real rank", "value": "2", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#center", "subject": "group:so-2-3-identity", "key": "center", "label": "Center", "value": "1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#pi1", "subject": "group:so-2-3-identity", "key": "pi1", "label": "Fundamental group π₁", "value": "Z × C₂", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.", "inputs": [ "Maximal compact subgroup: SO(2) × SO(3)", "π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z", "For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#compact", "subject": "group:so-2-3-identity", "key": "compact", "label": "Compact", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#connected", "subject": "group:so-2-3-identity", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#abelian", "subject": "group:so-2-3-identity", "key": "abelian", "label": "Abelian", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#simple-algebra", "subject": "group:so-2-3-identity", "key": "simple-algebra", "label": "Simple real Lie algebra", "value": "Yes", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" }, { "source": "conrad", "locator": "Lecture 25 and Appendix T: root systems and classical root computations" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#global-form", "subject": "group:so-2-3-identity", "key": "global-form", "label": "Global form", "value": "Sp(4,R)/{±I}", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic double cover of SO(3,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#lie-algebra", "subject": "group:so-2-3-identity", "key": "lie-algebra", "label": "Real Lie algebra", "value": "sp(4, R) ≅ so(2,3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#maximal-compact", "subject": "group:so-2-3-identity", "key": "maximal-compact", "label": "Maximal compact subgroup", "value": "SO(2) × SO(3)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#topology-1", "subject": "group:so-2-3-identity", "key": "topology-1", "label": "Global structure and topology", "value": "Its maximal compact subgroup is SO(2) × SO(3), giving π₁ = Z × C₂. The connected spin cover is Sp(4,R), whose fundamental group is Z. Neither the word Spin nor this double cover makes that covering group simply connected.", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic double cover of SO(3,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#representations-1", "subject": "group:so-2-3-identity", "key": "representations-1", "label": "Representation category and descent", "value": "The vector 5 is an ordinary representation of SO₀(2,3). The symplectic 4 belongs to its connected spin cover Sp(4,R), since the covering kernel acts by minus the identity.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8: four-dimensional defining module and five-dimensional orthogonal action" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#relation-1", "subject": "group:so-2-3-identity", "key": "relation-1", "label": "Real Lie algebra", "value": "algebra:sp-4-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#relation-2", "subject": "group:so-2-3-identity", "key": "relation-2", "label": "Real-form context", "value": "real-form:sp-4-r", "evidence": [ { "source": "etingof-real-forms", "locator": "Sections 41.2–41.4: compact, split, and classical real forms" } ], "assumptions": [], "relationship": "has-real-form", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#relation-3", "subject": "group:so-2-3-identity", "key": "relation-3", "label": "Explicit homomorphism", "value": "map:sp-4-r-to-so-2-3", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic double cover of SO(3,2)" } ], "assumptions": [], "relationship": "has-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:so-2-3-identity#relation-4", "subject": "group:so-2-3-identity", "key": "relation-4", "label": "Specified connected covering group", "value": "group:sp-4-r", "evidence": [ { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "garrett-isogenies", "locator": "Section 2.8: real symplectic double cover of SO(3,2)" } ], "assumptions": [], "relationship": "covered-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#definition", "subject": "map:su-2-to-so-3", "key": "definition", "label": "Definition", "value": "A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#domain", "subject": "map:su-2-to-so-3", "key": "domain", "label": "Domain", "value": "su-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#codomain", "subject": "map:su-2-to-so-3", "key": "codomain", "label": "Codomain", "value": "so-3", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#formula", "subject": "map:su-2-to-so-3", "key": "formula", "label": "Specified homomorphism", "value": "A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#kernel", "subject": "map:su-2-to-so-3", "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#image", "subject": "map:su-2-to-so-3", "key": "image", "label": "Image", "value": "SO(3)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#injective", "subject": "map:su-2-to-so-3", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#surjective", "subject": "map:su-2-to-so-3", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#covering-degree", "subject": "map:su-2-to-so-3", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#meaning-1", "subject": "map:su-2-to-so-3", "key": "meaning-1", "label": "Global and representation consequences", "value": "This is the universal covering map. A spin-j SU(2) representation descends along it exactly when 2j is even. With compatible axis conventions, diag(eⁱθ,e⁻ⁱθ) maps to a rotation through 2θ.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#meaning-2", "subject": "map:su-2-to-so-3", "key": "meaning-2", "label": "Global and representation consequences", "value": "The torus angle in SU(2) has period 2π, but the image rotation has angle 2θ. Tracking that factor is necessary when comparing the two spherical quotient fibrations.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "garrett-isogenies", "locator": "Section 2.1: adjoint rotation action" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#relation-1", "subject": "map:su-2-to-so-3", "key": "relation-1", "label": "Domain group", "value": "group:su-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-so-3#relation-2", "subject": "map:su-2-to-so-3", "key": "relation-2", "label": "Codomain group", "value": "group:so-3", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.1: SU(2) → SO(3), construction and kernel" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#definition", "subject": "map:su-3-to-psu-3", "key": "definition", "label": "Definition", "value": "A ↦ its coset in SU(3)/C₃.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#domain", "subject": "map:su-3-to-psu-3", "key": "domain", "label": "Domain", "value": "su-3", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#codomain", "subject": "map:su-3-to-psu-3", "key": "codomain", "label": "Codomain", "value": "psu-3", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#formula", "subject": "map:su-3-to-psu-3", "key": "formula", "label": "Specified homomorphism", "value": "A ↦ its coset in SU(3)/C₃.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#kernel", "subject": "map:su-3-to-psu-3", "key": "kernel", "label": "Kernel", "value": "{exp(2πik/3)I : k = 0,1,2} ≅ C₃", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#image", "subject": "map:su-3-to-psu-3", "key": "image", "label": "Image", "value": "PSU(3)", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#injective", "subject": "map:su-3-to-psu-3", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#surjective", "subject": "map:su-3-to-psu-3", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#covering-degree", "subject": "map:su-3-to-psu-3", "key": "covering-degree", "label": "Covering degree", "value": "3", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#meaning-1", "subject": "map:su-3-to-psu-3", "key": "meaning-1", "label": "Global and representation consequences", "value": "The defining 3 is not a representation of PSU(3). A highest weight (p,q) descends precisely when p + 2q ≡ 0 mod 3.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#relation-1", "subject": "map:su-3-to-psu-3", "key": "relation-1", "label": "Domain group", "value": "group:su-3", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-to-psu-3#relation-2", "subject": "map:su-3-to-psu-3", "key": "relation-2", "label": "Codomain group", "value": "group:psu-3", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#definition", "subject": "map:su-2-to-su-3-regular", "key": "definition", "label": "Definition", "value": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#domain", "subject": "map:su-2-to-su-3-regular", "key": "domain", "label": "Domain", "value": "su-2", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#codomain", "subject": "map:su-2-to-su-3-regular", "key": "codomain", "label": "Codomain", "value": "su-3", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#formula", "subject": "map:su-2-to-su-3-regular", "key": "formula", "label": "Specified homomorphism", "value": "A ↦ diag(A,1). The first two coordinates form the defining SU(2) doublet and the third coordinate is fixed.", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#kernel", "subject": "map:su-2-to-su-3-regular", "key": "kernel", "label": "Kernel", "value": "{I}", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#image", "subject": "map:su-2-to-su-3-regular", "key": "image", "label": "Image", "value": "{diag(A,1) : A ∈ SU(2)} ≅ SU(2)", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#injective", "subject": "map:su-2-to-su-3-regular", "key": "injective", "label": "Injective", "value": "Yes", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#surjective", "subject": "map:su-2-to-su-3-regular", "key": "surjective", "label": "Surjective", "value": "No", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#meaning-1", "subject": "map:su-2-to-su-3-regular", "key": "meaning-1", "label": "Global and representation consequences", "value": "The SU(3) fundamental branches as 3 → 2 ⊕ 1. The adjoint branches as 8 → 3 ⊕ 2 ⊕ 2 ⊕ 1.", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#meaning-2", "subject": "map:su-2-to-su-3-regular", "key": "meaning-2", "label": "Global and representation consequences", "value": "For the adjoint rule, restrict End(C³) and remove its scalar summand. End(2 ⊕ 1) decomposes as (3 ⊕ 1) ⊕ 2 ⊕ 2 ⊕ 1, giving the stated eight-dimensional result after subtracting one singlet.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" }, { "source": "haber", "locator": "Problem 5: regular inclusion" } ], "derivation": { "method": "Restrict End(3) − 1 using the specified block inclusion and the SU(2) tensor rule 2⊗2=3⊕1.", "inputs": [ "3 restricts to 2⊕1", "Adjoint SU(3) = End(3) minus scalars" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#relation-1", "subject": "map:su-2-to-su-3-regular", "key": "relation-1", "label": "Domain group", "value": "group:su-2", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-regular#relation-2", "subject": "map:su-2-to-su-3-regular", "key": "relation-2", "label": "Codomain group", "value": "group:su-3", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: block SU(2) subgroup of SU(3)" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#definition", "subject": "map:su-2-to-su-3-principal", "key": "definition", "label": "Definition", "value": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#domain", "subject": "map:su-2-to-su-3-principal", "key": "domain", "label": "Domain", "value": "su-2", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#codomain", "subject": "map:su-2-to-su-3-principal", "key": "codomain", "label": "Codomain", "value": "su-3", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#formula", "subject": "map:su-2-to-su-3-principal", "key": "formula", "label": "Specified homomorphism", "value": "A ↦ Sym²(A) on Sym²(C²), with a unitary basis fixed. A real orthonormal form identifies its image with SO(3).", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#kernel", "subject": "map:su-2-to-su-3-principal", "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#image", "subject": "map:su-2-to-su-3-principal", "key": "image", "label": "Image", "value": "An SU(3)-conjugate of the real SO(3) subgroup", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#injective", "subject": "map:su-2-to-su-3-principal", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#surjective", "subject": "map:su-2-to-su-3-principal", "key": "surjective", "label": "Surjective", "value": "No", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#meaning-1", "subject": "map:su-2-to-su-3-principal", "key": "meaning-1", "label": "Global and representation consequences", "value": "The fundamental branches as 3 → 3, and the adjoint as 8 → 3 ⊕ 5. This is a Lie-algebra embedding, but the group homomorphism is not injective.", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#meaning-2", "subject": "map:su-2-to-su-3-principal", "key": "meaning-2", "label": "Global and representation consequences", "value": "The adjoint rule follows from 3 ⊗ 3 = 1 ⊕ 3 ⊕ 5 after removing the scalar endomorphism. Every surviving summand has integer spin, consistent with the central kernel.", "evidence": [ { "source": "conrad", "locator": "Lectures 27–28: highest-weight classification and fundamental representations" } ], "derivation": { "method": "Use the SU(2) spin-one tensor decomposition and Adj(SU(3)) = End(3) − 1.", "inputs": [ "The defining SU(3) representation restricts to spin 1", "Spin 1 ⊗ spin 1 = spin 0 ⊕ spin 1 ⊕ spin 2" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#relation-1", "subject": "map:su-2-to-su-3-principal", "key": "relation-1", "label": "Domain group", "value": "group:su-2", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-to-su-3-principal#relation-2", "subject": "map:su-2-to-su-3-principal", "key": "relation-2", "label": "Codomain group", "value": "group:su-3", "evidence": [ { "source": "haber", "locator": "Problem 5, pages 14–15: the irreducible three-dimensional SU(2) action and SO(3) image" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#definition", "subject": "map:u-1-to-su-2", "key": "definition", "label": "Definition", "value": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#domain", "subject": "map:u-1-to-su-2", "key": "domain", "label": "Domain", "value": "u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#codomain", "subject": "map:u-1-to-su-2", "key": "codomain", "label": "Codomain", "value": "su-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#formula", "subject": "map:u-1-to-su-2", "key": "formula", "label": "Specified homomorphism", "value": "eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ), with θ identified modulo 2π.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#kernel", "subject": "map:u-1-to-su-2", "key": "kernel", "label": "Kernel", "value": "{1}", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#image", "subject": "map:u-1-to-su-2", "key": "image", "label": "Image", "value": "{diag(z,z⁻¹) : |z|=1}", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#injective", "subject": "map:u-1-to-su-2", "key": "injective", "label": "Injective", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#surjective", "subject": "map:u-1-to-su-2", "key": "surjective", "label": "Surjective", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#meaning-1", "subject": "map:u-1-to-su-2", "key": "meaning-1", "label": "Global and representation consequences", "value": "The defining doublet has U(1) weights +1 and −1 in this convention. Its homogeneous quotient is SU(2)/U(1) ≅ S².", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#meaning-2", "subject": "map:u-1-to-su-2", "key": "meaning-2", "label": "Global and representation consequences", "value": "The induced map π₁(U(1)) → π₁(SU(2)) is zero. In the quotient fibration, the boundary map π₂(S²) → π₁(U(1)) is multiplication by +1 when the sphere generator is oriented to match this fiber generator.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Apply the exact sequence to U(1) → SU(2) → S², using π₂(S³)=π₁(S³)=0.", "inputs": [ "SU(2) is S³", "π₁(U(1))=Z", "Fiber loop θ:0→2π", "Choice of positively compatible sphere generator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#relation-1", "subject": "map:u-1-to-su-2", "key": "relation-1", "label": "Domain group", "value": "group:u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#relation-2", "subject": "map:u-1-to-su-2", "key": "relation-2", "label": "Codomain group", "value": "group:su-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-su-2#relation-3", "subject": "map:u-1-to-su-2", "key": "relation-3", "label": "Associated homogeneous quotient", "value": "space:su-2-over-u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "defines-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#definition", "subject": "map:u-1-to-so-3", "key": "definition", "label": "Definition", "value": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ modulo 2π.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#domain", "subject": "map:u-1-to-so-3", "key": "domain", "label": "Domain", "value": "u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#codomain", "subject": "map:u-1-to-so-3", "key": "codomain", "label": "Codomain", "value": "so-3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#formula", "subject": "map:u-1-to-so-3", "key": "formula", "label": "Specified homomorphism", "value": "eⁱθ ↦ diag(R(θ),1), where R(θ) has rows (cos θ,−sin θ) and (sin θ,cos θ), with θ modulo 2π.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#kernel", "subject": "map:u-1-to-so-3", "key": "kernel", "label": "Kernel", "value": "{1}", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#image", "subject": "map:u-1-to-so-3", "key": "image", "label": "Image", "value": "Rotations fixing a chosen oriented axis, isomorphic to SO(2)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#injective", "subject": "map:u-1-to-so-3", "key": "injective", "label": "Injective", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#surjective", "subject": "map:u-1-to-so-3", "key": "surjective", "label": "Surjective", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#meaning-1", "subject": "map:u-1-to-so-3", "key": "meaning-1", "label": "Global and representation consequences", "value": "This uses the actual spatial rotation angle θ. The quotient SO(3)/SO(2) is S². The inclusion sends a winding number to its parity in π₁(SO(3)) = C₂.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#meaning-2", "subject": "map:u-1-to-so-3", "key": "meaning-2", "label": "Global and representation consequences", "value": "A 2π rotation lifts to a path from I to −I in SU(2), whereas a 4π rotation lifts to a loop. Therefore the inclusion induces reduction modulo two, and the boundary map π₂(S²) → π₁(SO(2)) is multiplication by +2 after choosing compatible orientations.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "The kernel of Z → C₂ is 2Z. Since π₂(SO(3))=0, the boundary map is injective with this image.", "inputs": [ "map:su-2-to-so-3", "SO(3) is RP³", "π₂(RP³)=π₂(S³)=0", "Fiber loop is one spatial 2π rotation" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#relation-1", "subject": "map:u-1-to-so-3", "key": "relation-1", "label": "Domain group", "value": "group:u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#relation-2", "subject": "map:u-1-to-so-3", "key": "relation-2", "label": "Codomain group", "value": "group:so-3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:u-1-to-so-3#relation-3", "subject": "map:u-1-to-so-3", "key": "relation-3", "label": "Associated homogeneous quotient", "value": "space:so-3-over-so-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "defines-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#definition", "subject": "map:spin-4-to-so-4", "key": "definition", "label": "Definition", "value": "(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#domain", "subject": "map:spin-4-to-so-4", "key": "domain", "label": "Domain", "value": "spin-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#codomain", "subject": "map:spin-4-to-so-4", "key": "codomain", "label": "Codomain", "value": "so-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#formula", "subject": "map:spin-4-to-so-4", "key": "formula", "label": "Specified homomorphism", "value": "(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#kernel", "subject": "map:spin-4-to-so-4", "key": "kernel", "label": "Kernel", "value": "{(1,1),(−1,−1)} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#image", "subject": "map:spin-4-to-so-4", "key": "image", "label": "Image", "value": "SO(4)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#injective", "subject": "map:spin-4-to-so-4", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#surjective", "subject": "map:spin-4-to-so-4", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#covering-degree", "subject": "map:spin-4-to-so-4", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#meaning-1", "subject": "map:spin-4-to-so-4", "key": "meaning-1", "label": "Global and representation consequences", "value": "The kernel is the diagonal order-two subgroup, not the full C₂ × C₂ center of Spin(4). Hence (jL,jR) descends exactly when jL + jR is an integer.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#relation-1", "subject": "map:spin-4-to-so-4", "key": "relation-1", "label": "Domain group", "value": "group:spin-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-4-to-so-4#relation-2", "subject": "map:spin-4-to-so-4", "key": "relation-2", "label": "Codomain group", "value": "group:so-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.3: quaternionic SO(4) covering construction" }, { "source": "haber", "locator": "Pages 7–8: SU(2) × SU(2) and SO(4)" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#definition", "subject": "map:su-4-to-so-6", "key": "definition", "label": "Definition", "value": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#domain", "subject": "map:su-4-to-so-6", "key": "domain", "label": "Domain", "value": "su-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#codomain", "subject": "map:su-4-to-so-6", "key": "codomain", "label": "Codomain", "value": "so-6", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#formula", "subject": "map:su-4-to-so-6", "key": "formula", "label": "Specified homomorphism", "value": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#kernel", "subject": "map:su-4-to-so-6", "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#image", "subject": "map:su-4-to-so-6", "key": "image", "label": "Image", "value": "SO(6)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#injective", "subject": "map:su-4-to-so-6", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#surjective", "subject": "map:su-4-to-so-6", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#covering-degree", "subject": "map:su-4-to-so-6", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#meaning-1", "subject": "map:su-4-to-so-6", "key": "meaning-1", "label": "Global and representation consequences", "value": "This identifies SU(4) with Spin(6). The central element iI acts by −I on the vector 6, while −I acts trivially; quotienting by the full C₄ center would give PSO(6).", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#relation-1", "subject": "map:su-4-to-so-6", "key": "relation-1", "label": "Domain group", "value": "group:su-4", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-so-6#relation-2", "subject": "map:su-4-to-so-6", "key": "relation-2", "label": "Codomain group", "value": "group:so-6", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#definition", "subject": "map:su-4-to-psu-4", "key": "definition", "label": "Definition", "value": "A ↦ its coset in SU(4)/C₄.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#domain", "subject": "map:su-4-to-psu-4", "key": "domain", "label": "Domain", "value": "su-4", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#codomain", "subject": "map:su-4-to-psu-4", "key": "codomain", "label": "Codomain", "value": "psu-4", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#formula", "subject": "map:su-4-to-psu-4", "key": "formula", "label": "Specified homomorphism", "value": "A ↦ its coset in SU(4)/C₄.", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#kernel", "subject": "map:su-4-to-psu-4", "key": "kernel", "label": "Kernel", "value": "{I,iI,−I,−iI} ≅ C₄", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#image", "subject": "map:su-4-to-psu-4", "key": "image", "label": "Image", "value": "PSU(4)", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#injective", "subject": "map:su-4-to-psu-4", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#surjective", "subject": "map:su-4-to-psu-4", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#covering-degree", "subject": "map:su-4-to-psu-4", "key": "covering-degree", "label": "Covering degree", "value": "4", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#meaning-1", "subject": "map:su-4-to-psu-4", "key": "meaning-1", "label": "Global and representation consequences", "value": "This is the full adjoint quotient. It has a different kernel from the vector-cover map SU(4) → SO(6).", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#relation-1", "subject": "map:su-4-to-psu-4", "key": "relation-1", "label": "Domain group", "value": "group:su-4", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-4-to-psu-4#relation-2", "subject": "map:su-4-to-psu-4", "key": "relation-2", "label": "Codomain group", "value": "group:psu-4", "evidence": [ { "source": "conrad-centers", "locator": "Pages 2–4: centers of simply connected groups by root-system type" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#definition", "subject": "map:sp-2-to-so-5", "key": "definition", "label": "Definition", "value": "A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#domain", "subject": "map:sp-2-to-so-5", "key": "domain", "label": "Domain", "value": "sp-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#codomain", "subject": "map:sp-2-to-so-5", "key": "codomain", "label": "Codomain", "value": "so-5", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#formula", "subject": "map:sp-2-to-so-5", "key": "formula", "label": "Specified homomorphism", "value": "A acts by X ↦ AXA† on the real five-dimensional space of traceless Hermitian quaternionic 2 × 2 matrices. The determinant is a definite quadratic form up to overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#kernel", "subject": "map:sp-2-to-so-5", "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#image", "subject": "map:sp-2-to-so-5", "key": "image", "label": "Image", "value": "SO(5)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#injective", "subject": "map:sp-2-to-so-5", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#surjective", "subject": "map:sp-2-to-so-5", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#covering-degree", "subject": "map:sp-2-to-so-5", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#meaning-1", "subject": "map:sp-2-to-so-5", "key": "meaning-1", "label": "Global and representation consequences", "value": "The quaternionic compact convention is essential: Sp(2) here is USp(4), not Sp(4,R). The map exhibits Sp(2) ≅ Spin(5).", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#relation-1", "subject": "map:sp-2-to-so-5", "key": "relation-1", "label": "Domain group", "value": "group:sp-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-2-to-so-5#relation-2", "subject": "map:sp-2-to-so-5", "key": "relation-2", "label": "Codomain group", "value": "group:so-5", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#definition", "subject": "map:spin-8-to-so-8", "key": "definition", "label": "Definition", "value": "g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8).", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#domain", "subject": "map:spin-8-to-so-8", "key": "domain", "label": "Domain", "value": "spin-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#codomain", "subject": "map:spin-8-to-so-8", "key": "codomain", "label": "Codomain", "value": "so-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#formula", "subject": "map:spin-8-to-so-8", "key": "formula", "label": "Specified homomorphism", "value": "g ↦ (v ↦ gvg⁻¹) on the copy of R⁸ in its Euclidean Clifford algebra, for g in Spin(8).", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#kernel", "subject": "map:spin-8-to-so-8", "key": "kernel", "label": "Kernel", "value": "{±1} ≅ C₂", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#image", "subject": "map:spin-8-to-so-8", "key": "image", "label": "Image", "value": "SO(8)", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#injective", "subject": "map:spin-8-to-so-8", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#surjective", "subject": "map:spin-8-to-so-8", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#covering-degree", "subject": "map:spin-8-to-so-8", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#meaning-1", "subject": "map:spin-8-to-so-8", "key": "meaning-1", "label": "Global and representation consequences", "value": "This specifies the vector-cover kernel inside the full C₂ × C₂ center. Triality changes the role of vector and half-spin modules and permutes the three order-two kernels.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#meaning-2", "subject": "map:spin-8-to-so-8", "key": "meaning-2", "label": "Global and representation consequences", "value": "Although the three quotients by order-two central subgroups are abstractly isomorphic, a representation descends according to the kernel of the specified map. The quotient name alone does not retain this choice.", "evidence": [ { "source": "conrad", "locator": "Remark T.2.1, printed page 262: D₄ triality and quotient groups" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#relation-1", "subject": "map:spin-8-to-so-8", "key": "relation-1", "label": "Domain group", "value": "group:spin-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-8-to-so-8#relation-2", "subject": "map:spin-8-to-so-8", "key": "relation-2", "label": "Codomain group", "value": "group:so-8", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#definition", "subject": "map:spin-10-to-so-10", "key": "definition", "label": "Definition", "value": "g ↦ (v ↦ gvg⁻¹) on R¹⁰ inside its Euclidean Clifford algebra.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#domain", "subject": "map:spin-10-to-so-10", "key": "domain", "label": "Domain", "value": "spin-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#codomain", "subject": "map:spin-10-to-so-10", "key": "codomain", "label": "Codomain", "value": "so-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#formula", "subject": "map:spin-10-to-so-10", "key": "formula", "label": "Specified homomorphism", "value": "g ↦ (v ↦ gvg⁻¹) on R¹⁰ inside its Euclidean Clifford algebra.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#kernel", "subject": "map:spin-10-to-so-10", "key": "kernel", "label": "Kernel", "value": "{±1} ≅ C₂", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#image", "subject": "map:spin-10-to-so-10", "key": "image", "label": "Image", "value": "SO(10)", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#injective", "subject": "map:spin-10-to-so-10", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#surjective", "subject": "map:spin-10-to-so-10", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#covering-degree", "subject": "map:spin-10-to-so-10", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#meaning-1", "subject": "map:spin-10-to-so-10", "key": "meaning-1", "label": "Global and representation consequences", "value": "The full center of Spin(10) is C₄, while this map has the unique order-two kernel. The half-spin 16s do not descend through it.", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#relation-1", "subject": "map:spin-10-to-so-10", "key": "relation-1", "label": "Domain group", "value": "group:spin-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:spin-10-to-so-10#relation-2", "subject": "map:spin-10-to-so-10", "key": "relation-2", "label": "Codomain group", "value": "group:so-10", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#definition", "subject": "map:sl-2-c-to-lorentz", "key": "definition", "label": "Definition", "value": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#domain", "subject": "map:sl-2-c-to-lorentz", "key": "domain", "label": "Domain", "value": "sl-2-c", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#codomain", "subject": "map:sl-2-c-to-lorentz", "key": "codomain", "label": "Codomain", "value": "so-1-3-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#formula", "subject": "map:sl-2-c-to-lorentz", "key": "formula", "label": "Specified homomorphism", "value": "Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#kernel", "subject": "map:sl-2-c-to-lorentz", "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#image", "subject": "map:sl-2-c-to-lorentz", "key": "image", "label": "Image", "value": "SO₀(1,3)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#injective", "subject": "map:sl-2-c-to-lorentz", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#surjective", "subject": "map:sl-2-c-to-lorentz", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#covering-degree", "subject": "map:sl-2-c-to-lorentz", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#meaning-1", "subject": "map:sl-2-c-to-lorentz", "key": "meaning-1", "label": "Global and representation consequences", "value": "The connected source maps onto the identity component of the Lorentz group. It does not cover time reversal or spatial parity components. SL(2,C) is simply connected as a real Lie group.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#meaning-2", "subject": "map:sl-2-c-to-lorentz", "key": "meaning-2", "label": "Global and representation consequences", "value": "Preservation of det X follows from det A = 1. If AXA† = X for every Hermitian X, then A is scalar and equals ±I. The derivative is an isomorphism of six-dimensional real Lie algebras; connectedness identifies the image as SO₀(1,3).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" } ], "derivation": { "method": "Check the determinant, central kernel, derivative, and identity component directly from the displayed action.", "inputs": [ "det A=1", "Hermitian 2×2 matrices form a real four-dimensional vector space", "SL(2,C) is connected" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#relation-1", "subject": "map:sl-2-c-to-lorentz", "key": "relation-1", "label": "Domain group", "value": "group:sl-2-c", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-c-to-lorentz#relation-2", "subject": "map:sl-2-c-to-lorentz", "key": "relation-2", "label": "Codomain group", "value": "group:so-1-3-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.4: SL(2,C) orthogonal covering of signature (3,1)" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#definition", "subject": "map:sl-2-r-to-psl-2-r", "key": "definition", "label": "Definition", "value": "A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#domain", "subject": "map:sl-2-r-to-psl-2-r", "key": "domain", "label": "Domain", "value": "sl-2-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#codomain", "subject": "map:sl-2-r-to-psl-2-r", "key": "codomain", "label": "Codomain", "value": "psl-2-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#formula", "subject": "map:sl-2-r-to-psl-2-r", "key": "formula", "label": "Specified homomorphism", "value": "A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#kernel", "subject": "map:sl-2-r-to-psl-2-r", "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#image", "subject": "map:sl-2-r-to-psl-2-r", "key": "image", "label": "Image", "value": "PSL(2,R) ≅ SO₀(2,1)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#injective", "subject": "map:sl-2-r-to-psl-2-r", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#surjective", "subject": "map:sl-2-r-to-psl-2-r", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#covering-degree", "subject": "map:sl-2-r-to-psl-2-r", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#meaning-1", "subject": "map:sl-2-r-to-psl-2-r", "key": "meaning-1", "label": "Global and representation consequences", "value": "Both groups have infinite cyclic fundamental group. With compatible circle generators, this double cover induces multiplication by two on π₁; neither group is the universal cover.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#relation-1", "subject": "map:sl-2-r-to-psl-2-r", "key": "relation-1", "label": "Domain group", "value": "group:sl-2-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sl-2-r-to-psl-2-r#relation-2", "subject": "map:sl-2-r-to-psl-2-r", "key": "relation-2", "label": "Codomain group", "value": "group:psl-2-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.2: SL(2,R) covering the identity component of SO(2,1)" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#definition", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "definition", "label": "Definition", "value": "Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#domain", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "domain", "label": "Domain", "value": "universal-cover-sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#codomain", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "codomain", "label": "Codomain", "value": "sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#formula", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "formula", "label": "Specified homomorphism", "value": "Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#kernel", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "kernel", "label": "Kernel", "value": "⟨z²⟩ ≅ Z, where z lifts a π rotation", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#image", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "image", "label": "Image", "value": "SL(2,R)", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#injective", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#surjective", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#covering-degree", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "covering-degree", "label": "Covering degree", "value": "Infinitely many sheets", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#meaning-1", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "meaning-1", "label": "Global and representation consequences", "value": "The center of the domain is ⟨z⟩. Its subgroup ⟨z²⟩ is the kernel of this projection; composing with SL(2,R) → PSL(2,R) enlarges the kernel to the full center.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#meaning-2", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "meaning-2", "label": "Global and representation consequences", "value": "Choose R(θ) as the standard planar rotation matrix inside SL(2,R). The path θ:0→π ends at −I and defines z; the path θ:0→2π is one fundamental loop in SL(2,R) and defines z².", "evidence": [ { "source": "etingof-global-forms", "locator": "Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)" }, { "source": "conrad", "locator": "Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients" } ], "derivation": { "method": "Lift the maximal compact rotation subgroup through the universal covering map.", "inputs": [ "SL(2,R) retracts to SO(2)", "π₁(SO(2)) is generated by θ:0→2π", "R(π)=−I" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#relation-1", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "relation-1", "label": "Domain group", "value": "group:universal-cover-sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:universal-sl-2-r-to-sl-2-r#relation-2", "subject": "map:universal-sl-2-r-to-sl-2-r", "key": "relation-2", "label": "Codomain group", "value": "group:sl-2-r", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 3.2: covering Lie groups" }, { "source": "brodzki-sl2-cover", "locator": "Section 1, page 2: universal cover and its infinite cyclic center" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#definition", "subject": "map:su-2-2-to-so-2-4", "key": "definition", "label": "Definition", "value": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#domain", "subject": "map:su-2-2-to-so-2-4", "key": "domain", "label": "Domain", "value": "su-2-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#codomain", "subject": "map:su-2-2-to-so-2-4", "key": "codomain", "label": "Codomain", "value": "so-2-4-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#formula", "subject": "map:su-2-2-to-so-2-4", "key": "formula", "label": "Specified homomorphism", "value": "Act by Λ²A on Λ²C⁴ and restrict to its SU(2,2)-invariant real six-dimensional form. The invariant wedge pairing has signature (2,4), up to reversing its overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#kernel", "subject": "map:su-2-2-to-so-2-4", "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#image", "subject": "map:su-2-2-to-so-2-4", "key": "image", "label": "Image", "value": "SO₀(2,4)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#injective", "subject": "map:su-2-2-to-so-2-4", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#surjective", "subject": "map:su-2-2-to-so-2-4", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#covering-degree", "subject": "map:su-2-2-to-so-2-4", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#meaning-1", "subject": "map:su-2-2-to-so-2-4", "key": "meaning-1", "label": "Global and representation consequences", "value": "This map identifies SU(2,2) with the connected spin double cover. Its source has π₁ = Z, so this is not a universal covering map. A further quotient of the target by {±I} is a different global group.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#relation-1", "subject": "map:su-2-2-to-so-2-4", "key": "relation-1", "label": "Domain group", "value": "group:su-2-2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-2-2-to-so-2-4#relation-2", "subject": "map:su-2-2-to-so-2-4", "key": "relation-2", "label": "Codomain group", "value": "group:so-2-4-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.11, pages 9–10: SU(2,2) → SO(4,2)" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#definition", "subject": "map:sp-4-r-to-so-2-3", "key": "definition", "label": "Definition", "value": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#domain", "subject": "map:sp-4-r-to-so-2-3", "key": "domain", "label": "Domain", "value": "sp-4-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#codomain", "subject": "map:sp-4-r-to-so-2-3", "key": "codomain", "label": "Codomain", "value": "so-2-3-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#formula", "subject": "map:sp-4-r-to-so-2-3", "key": "formula", "label": "Specified homomorphism", "value": "For a real symplectic vector space V of dimension 4, act on the five-dimensional primitive summand of Λ²V. Its wedge pairing is a nondegenerate symmetric form of signature (2,3), up to overall sign.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#kernel", "subject": "map:sp-4-r-to-so-2-3", "key": "kernel", "label": "Kernel", "value": "{±I} ≅ C₂", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#image", "subject": "map:sp-4-r-to-so-2-3", "key": "image", "label": "Image", "value": "SO₀(2,3)", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#injective", "subject": "map:sp-4-r-to-so-2-3", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#surjective", "subject": "map:sp-4-r-to-so-2-3", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#covering-degree", "subject": "map:sp-4-r-to-so-2-3", "key": "covering-degree", "label": "Covering degree", "value": "2", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#meaning-1", "subject": "map:sp-4-r-to-so-2-3", "key": "meaning-1", "label": "Global and representation consequences", "value": "The source has maximal compact U(2) and fundamental group Z; the target has maximal compact SO(2) × SO(3) and fundamental group Z × C₂. The covering is double but is not universal.", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" }, { "source": "etingof-global-forms", "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#meaning-2", "subject": "map:sp-4-r-to-so-2-3", "key": "meaning-2", "label": "Global and representation consequences", "value": "On maximal compact subgroups, the covering may be written U(2) → SO(2) × SO(3) using the determinant and the projective SU(2) action. A generator of π₁(U(2)) maps to (1,1 mod 2), an index-two subgroup of Z × C₂.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" }, { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "knapp-structure", "locator": "Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316" } ], "derivation": { "method": "Represent a generator by diag(eⁱᵗ,1), 0≤t≤2π. Its determinant winds once and its SU(2) lift ends at −I, giving the nontrivial SO(3) loop.", "inputs": [ "U(2)=(SU(2)×U(1))/diagonal C₂", "SU(2)→SO(3) has kernel {±I}", "The determinant map induces π₁(U(2))≅Z" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#relation-1", "subject": "map:sp-4-r-to-so-2-3", "key": "relation-1", "label": "Domain group", "value": "group:sp-4-r", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:sp-4-r-to-so-2-3#relation-2", "subject": "map:sp-4-r-to-so-2-3", "key": "relation-2", "label": "Codomain group", "value": "group:so-2-3-identity", "evidence": [ { "source": "garrett-isogenies", "locator": "Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R)" }, { "source": "etingof-lie-notes", "locator": "Section 31.1, printed page 165: the primitive exterior-square symplectic module" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#definition", "subject": "space:su-2-over-u-1", "key": "definition", "label": "Definition", "value": "The right-coset homogeneous space of SU(2) by the embedded torus eⁱθ ↦ diag(eⁱθ,e⁻ⁱθ).", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#dimension", "subject": "space:su-2-over-u-1", "key": "dimension", "label": "Real dimension", "value": "2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#manifold", "subject": "space:su-2-over-u-1", "key": "manifold", "label": "Manifold", "value": "S²", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#pi1", "subject": "space:su-2-over-u-1", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#pi2", "subject": "space:su-2-over-u-1", "key": "pi2", "label": "Second homotopy group π₂", "value": "Z", "evidence": [ { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#fiber-period", "subject": "space:su-2-over-u-1", "key": "fiber-period", "label": "Fiber angle period", "value": "2π", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#boundary-map", "subject": "space:su-2-over-u-1", "key": "boundary-map", "label": "Boundary π₂(S²) → π₁(U(1))", "value": "n ↦ n", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Exactness of 0 → π₂(S²) → π₁(U(1)) → 0 forces the boundary to be an isomorphism; orientation fixes its positive sign.", "inputs": [ "SU(2)=S³", "π₂(S³)=0", "π₁(S³)=0", "π₁(U(1))=Z" ] }, "assumptions": [ "The U(1) generator is θ:0→2π in diag(eⁱθ,e⁻ⁱθ).", "Choose the sphere generator so its boundary is the positive fiber generator." ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#sequence-1", "subject": "space:su-2-over-u-1", "key": "sequence-1", "label": "The exact sequence with generators", "value": "The relevant segment is 0 → Z → Z → 0. The two displayed Z groups are respectively π₂(S²) and π₁(U(1)). With the generator convention stated here, the first nonzero arrow is multiplication by +1.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Apply the fibration homotopy sequence to U(1) → S³ → S².", "inputs": [ "map:u-1-to-su-2", "π₂(S³)=π₁(S³)=0" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#sequence-2", "subject": "space:su-2-over-u-1", "key": "sequence-2", "label": "The exact sequence with generators", "value": "The same base manifold occurs for SO(3)/SO(2), but the embedding and the total group change the boundary homomorphism. The integer here is a fiber winding; converting it to a physical magnetic charge also requires the generator and coupling convention.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#relation-1", "subject": "space:su-2-over-u-1", "key": "relation-1", "label": "Total group", "value": "group:su-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "quotient-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#relation-2", "subject": "space:su-2-over-u-1", "key": "relation-2", "label": "Embedded torus", "value": "group:u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "has-isotropy", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#relation-3", "subject": "space:su-2-over-u-1", "key": "relation-3", "label": "Normalized subgroup map", "value": "map:u-1-to-su-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "defined-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:su-2-over-u-1#relation-4", "subject": "space:su-2-over-u-1", "key": "relation-4", "label": "Same S², different fibration", "value": "space:so-3-over-so-2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "same-manifold-as", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#definition", "subject": "space:so-3-over-so-2", "key": "definition", "label": "Definition", "value": "The homogeneous space of rotations modulo the subgroup fixing a chosen axis, with SO(2) identified with U(1) by the actual spatial rotation angle.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#dimension", "subject": "space:so-3-over-so-2", "key": "dimension", "label": "Real dimension", "value": "2", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#manifold", "subject": "space:so-3-over-so-2", "key": "manifold", "label": "Manifold", "value": "S²", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#pi1", "subject": "space:so-3-over-so-2", "key": "pi1", "label": "Fundamental group π₁", "value": "0", "evidence": [ { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#pi2", "subject": "space:so-3-over-so-2", "key": "pi2", "label": "Second homotopy group π₂", "value": "Z", "evidence": [ { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#fiber-period", "subject": "space:so-3-over-so-2", "key": "fiber-period", "label": "Physical rotation angle period", "value": "2π", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#inclusion-on-pi1", "subject": "space:so-3-over-so-2", "key": "inclusion-on-pi1", "label": "Inclusion π₁(SO(2)) → π₁(SO(3))", "value": "n ↦ n mod 2", "evidence": [ { "source": "conrad", "locator": "Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3" }, { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "derivation": { "method": "A 2π spatial rotation lifts from I to −I in SU(2), while a 4π rotation closes.", "inputs": [ "map:su-2-to-so-3", "Fiber angle is the spatial angle θ with period 2π" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#boundary-map", "subject": "space:so-3-over-so-2", "key": "boundary-map", "label": "Boundary π₂(S²) → π₁(SO(2))", "value": "n ↦ 2n", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Exactness identifies the boundary image with the kernel 2Z of Z→C₂. The boundary is injective because π₂(SO(3))=0.", "inputs": [ "SO(3)=RP³", "π₂(RP³)=π₂(S³)=0", "π₁(SO(3))=C₂", "Inclusion sends winding to its parity" ] }, "assumptions": [ "The fiber generator is one positive spatial 2π rotation.", "Choose the sphere generator to make the boundary coefficient positive." ], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#sequence-1", "subject": "space:so-3-over-so-2", "key": "sequence-1", "label": "The exact sequence with generators", "value": "The relevant segment is 0 → Z → Z → C₂ → 0. The arrow between the two infinite cyclic groups is multiplication by +2, and the following arrow reduces an integer modulo two.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" }, { "source": "hatcher-algebraic-topology", "locator": "Section 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375" } ], "derivation": { "method": "Use the fibration SO(2) → SO(3) → S² and the induced map on fundamental groups.", "inputs": [ "map:u-1-to-so-3", "π₂(SO(3))=0", "π₁(SO(3))=C₂", "π₁(S²)=0" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#sequence-2", "subject": "space:so-3-over-so-2", "key": "sequence-2", "label": "The exact sequence with generators", "value": "The subgroup record is U(1), used through the specified isomorphism U(1) ≅ SO(2). This avoids creating a second abstract circle-group record while retaining its concrete rotational action.", "evidence": [ { "source": "etingof-lie-notes", "locator": "Sections 6.1–6.3: classical matrix groups and their Lie algebras" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#relation-1", "subject": "space:so-3-over-so-2", "key": "relation-1", "label": "Total group", "value": "group:so-3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "quotient-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#relation-2", "subject": "space:so-3-over-so-2", "key": "relation-2", "label": "Circle realized as SO(2)", "value": "group:u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "has-isotropy", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#relation-3", "subject": "space:so-3-over-so-2", "key": "relation-3", "label": "Normalized rotational subgroup map", "value": "map:u-1-to-so-3", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "defined-by", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "space:so-3-over-so-2#relation-4", "subject": "space:so-3-over-so-2", "key": "relation-4", "label": "Same S², different fibration", "value": "space:su-2-over-u-1", "evidence": [ { "source": "etingof-lie-notes", "locator": "Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³" } ], "assumptions": [], "relationship": "same-manifold-as", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#definition", "subject": "representation:c-3-1", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1 of C₃, specified by its character on the named conjugacy classes. Trivial representation", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#dimension", "subject": "representation:c-3-1", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#field", "subject": "representation:c-3-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#reality", "subject": "representation:c-3-1", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#indicator", "subject": "representation:c-3-1", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#conjugate", "subject": "representation:c-3-1", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#representations-1", "subject": "representation:c-3-1", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. The two nonidentity classes are the chosen generator r and its square.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#physics-1", "subject": "representation:c-3-1", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#conventions-1", "subject": "representation:c-3-1", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-1#relation-1", "subject": "representation:c-3-1", "key": "relation-1", "label": "Representation domain", "value": "group:c-3", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#definition", "subject": "representation:c-3-chi", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled χ of C₃, specified by its character on the named conjugacy classes. The chosen generator acts by omega", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#dimension", "subject": "representation:c-3-chi", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#field", "subject": "representation:c-3-chi", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#reality", "subject": "representation:c-3-chi", "key": "reality", "label": "Frobenius–Schur type", "value": "Complex", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#indicator", "subject": "representation:c-3-chi", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "0", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#conjugate", "subject": "representation:c-3-chi", "key": "conjugate", "label": "Complex conjugate irrep", "value": "χ²", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#representations-1", "subject": "representation:c-3-chi", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. The two nonidentity classes are the chosen generator r and its square.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#physics-1", "subject": "representation:c-3-chi", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#conventions-1", "subject": "representation:c-3-chi", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; complex describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#relation-1", "subject": "representation:c-3-chi", "key": "relation-1", "label": "Representation domain", "value": "group:c-3", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi#relation-2", "subject": "representation:c-3-chi", "key": "relation-2", "label": "Complex conjugate representation", "value": "representation:c-3-chi2", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "conjugate", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#definition", "subject": "representation:c-3-chi2", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled χ² of C₃, specified by its character on the named conjugacy classes. The chosen generator acts by omega squared", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#dimension", "subject": "representation:c-3-chi2", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#field", "subject": "representation:c-3-chi2", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#reality", "subject": "representation:c-3-chi2", "key": "reality", "label": "Frobenius–Schur type", "value": "Complex", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#indicator", "subject": "representation:c-3-chi2", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "0", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#conjugate", "subject": "representation:c-3-chi2", "key": "conjugate", "label": "Complex conjugate irrep", "value": "χ", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#representations-1", "subject": "representation:c-3-chi2", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. The two nonidentity classes are the chosen generator r and its square.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#physics-1", "subject": "representation:c-3-chi2", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#conventions-1", "subject": "representation:c-3-chi2", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; complex describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#relation-1", "subject": "representation:c-3-chi2", "key": "relation-1", "label": "Representation domain", "value": "group:c-3", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:c-3-chi2#relation-2", "subject": "representation:c-3-chi2", "key": "relation-2", "label": "Complex conjugate representation", "value": "representation:c-3-chi", "evidence": [ { "source": "groupnames-c-3", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of C₃; the irrep χ² is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "conjugate", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#definition", "subject": "representation:s-3-1", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1 of S₃, specified by its character on the named conjugacy classes. Trivial representation", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#dimension", "subject": "representation:s-3-1", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#field", "subject": "representation:s-3-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#reality", "subject": "representation:s-3-1", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#indicator", "subject": "representation:s-3-1", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#conjugate", "subject": "representation:s-3-1", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#representations-1", "subject": "representation:s-3-1", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, transpositions, and three-cycles.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#physics-1", "subject": "representation:s-3-1", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#conventions-1", "subject": "representation:s-3-1", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-1#relation-1", "subject": "representation:s-3-1", "key": "relation-1", "label": "Representation domain", "value": "group:s-3", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#definition", "subject": "representation:s-3-sgn", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled sgn of S₃, specified by its character on the named conjugacy classes. Sign of a permutation", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#dimension", "subject": "representation:s-3-sgn", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#field", "subject": "representation:s-3-sgn", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#reality", "subject": "representation:s-3-sgn", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#indicator", "subject": "representation:s-3-sgn", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#conjugate", "subject": "representation:s-3-sgn", "key": "conjugate", "label": "Complex conjugate irrep", "value": "sgn", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#representations-1", "subject": "representation:s-3-sgn", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, transpositions, and three-cycles.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#physics-1", "subject": "representation:s-3-sgn", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#conventions-1", "subject": "representation:s-3-sgn", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-sgn#relation-1", "subject": "representation:s-3-sgn", "key": "relation-1", "label": "Representation domain", "value": "group:s-3", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#definition", "subject": "representation:s-3-2", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 2 of S₃, specified by its character on the named conjugacy classes. Standard representation: permutation representation minus the invariant line", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#dimension", "subject": "representation:s-3-2", "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#field", "subject": "representation:s-3-2", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#reality", "subject": "representation:s-3-2", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#indicator", "subject": "representation:s-3-2", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#conjugate", "subject": "representation:s-3-2", "key": "conjugate", "label": "Complex conjugate irrep", "value": "2", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#representations-1", "subject": "representation:s-3-2", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, transpositions, and three-cycles.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#physics-1", "subject": "representation:s-3-2", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#conventions-1", "subject": "representation:s-3-2", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-3-2#relation-1", "subject": "representation:s-3-2", "key": "relation-1", "label": "Representation domain", "value": "group:s-3", "evidence": [ { "source": "groupnames-s-3", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₃; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#definition", "subject": "representation:d-8-1", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1 of D₈, specified by its character on the named conjugacy classes. Trivial representation", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#dimension", "subject": "representation:d-8-1", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#field", "subject": "representation:d-8-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#reality", "subject": "representation:d-8-1", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#indicator", "subject": "representation:d-8-1", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#conjugate", "subject": "representation:d-8-1", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#representations-1", "subject": "representation:d-8-1", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#physics-1", "subject": "representation:d-8-1", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#conventions-1", "subject": "representation:d-8-1", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-1#relation-1", "subject": "representation:d-8-1", "key": "relation-1", "label": "Representation domain", "value": "group:d-8", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#definition", "subject": "representation:d-8-a", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled a of D₈, specified by its character on the named conjugacy classes. r maps to +1 and s to -1; determinant of the doublet", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#dimension", "subject": "representation:d-8-a", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#field", "subject": "representation:d-8-a", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#reality", "subject": "representation:d-8-a", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#indicator", "subject": "representation:d-8-a", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#conjugate", "subject": "representation:d-8-a", "key": "conjugate", "label": "Complex conjugate irrep", "value": "a", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#representations-1", "subject": "representation:d-8-a", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#physics-1", "subject": "representation:d-8-a", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#conventions-1", "subject": "representation:d-8-a", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-a#relation-1", "subject": "representation:d-8-a", "key": "relation-1", "label": "Representation domain", "value": "group:d-8", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#definition", "subject": "representation:d-8-b", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled b of D₈, specified by its character on the named conjugacy classes. r maps to -1 and s to +1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#dimension", "subject": "representation:d-8-b", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#field", "subject": "representation:d-8-b", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#reality", "subject": "representation:d-8-b", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#indicator", "subject": "representation:d-8-b", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#conjugate", "subject": "representation:d-8-b", "key": "conjugate", "label": "Complex conjugate irrep", "value": "b", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#representations-1", "subject": "representation:d-8-b", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#physics-1", "subject": "representation:d-8-b", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#conventions-1", "subject": "representation:d-8-b", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-b#relation-1", "subject": "representation:d-8-b", "key": "relation-1", "label": "Representation domain", "value": "group:d-8", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#definition", "subject": "representation:d-8-c", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled c of D₈, specified by its character on the named conjugacy classes. r maps to -1 and s to -1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#dimension", "subject": "representation:d-8-c", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#field", "subject": "representation:d-8-c", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#reality", "subject": "representation:d-8-c", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#indicator", "subject": "representation:d-8-c", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#conjugate", "subject": "representation:d-8-c", "key": "conjugate", "label": "Complex conjugate irrep", "value": "c", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#representations-1", "subject": "representation:d-8-c", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#physics-1", "subject": "representation:d-8-c", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#conventions-1", "subject": "representation:d-8-c", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-c#relation-1", "subject": "representation:d-8-c", "key": "relation-1", "label": "Representation domain", "value": "group:d-8", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#definition", "subject": "representation:d-8-2", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 2 of D₈, specified by its character on the named conjugacy classes. Faithful real-type geometric doublet", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#dimension", "subject": "representation:d-8-2", "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#field", "subject": "representation:d-8-2", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#reality", "subject": "representation:d-8-2", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#indicator", "subject": "representation:d-8-2", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#conjugate", "subject": "representation:d-8-2", "key": "conjugate", "label": "Complex conjugate irrep", "value": "2", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#representations-1", "subject": "representation:d-8-2", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, r², {r,r³}, {s,r²s}, {rs,r³s}. The character a is the determinant of the geometric doublet.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#physics-1", "subject": "representation:d-8-2", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#conventions-1", "subject": "representation:d-8-2", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:d-8-2#relation-1", "subject": "representation:d-8-2", "key": "relation-1", "label": "Representation domain", "value": "group:d-8", "evidence": [ { "source": "groupnames-d-8", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of D₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#definition", "subject": "representation:q-8-1", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1 of Q₈, specified by its character on the named conjugacy classes. Trivial representation", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#dimension", "subject": "representation:q-8-1", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#field", "subject": "representation:q-8-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#reality", "subject": "representation:q-8-1", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#indicator", "subject": "representation:q-8-1", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#conjugate", "subject": "representation:q-8-1", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#representations-1", "subject": "representation:q-8-1", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#physics-1", "subject": "representation:q-8-1", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#conventions-1", "subject": "representation:q-8-1", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-1#relation-1", "subject": "representation:q-8-1", "key": "relation-1", "label": "Representation domain", "value": "group:q-8", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#definition", "subject": "representation:q-8-a", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled a of Q₈, specified by its character on the named conjugacy classes. i maps to +1 and j to -1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#dimension", "subject": "representation:q-8-a", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#field", "subject": "representation:q-8-a", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#reality", "subject": "representation:q-8-a", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#indicator", "subject": "representation:q-8-a", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#conjugate", "subject": "representation:q-8-a", "key": "conjugate", "label": "Complex conjugate irrep", "value": "a", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#representations-1", "subject": "representation:q-8-a", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#physics-1", "subject": "representation:q-8-a", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#conventions-1", "subject": "representation:q-8-a", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-a#relation-1", "subject": "representation:q-8-a", "key": "relation-1", "label": "Representation domain", "value": "group:q-8", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep a is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#definition", "subject": "representation:q-8-b", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled b of Q₈, specified by its character on the named conjugacy classes. i maps to -1 and j to +1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#dimension", "subject": "representation:q-8-b", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#field", "subject": "representation:q-8-b", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#reality", "subject": "representation:q-8-b", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#indicator", "subject": "representation:q-8-b", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#conjugate", "subject": "representation:q-8-b", "key": "conjugate", "label": "Complex conjugate irrep", "value": "b", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#representations-1", "subject": "representation:q-8-b", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#physics-1", "subject": "representation:q-8-b", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#conventions-1", "subject": "representation:q-8-b", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-b#relation-1", "subject": "representation:q-8-b", "key": "relation-1", "label": "Representation domain", "value": "group:q-8", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep b is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#definition", "subject": "representation:q-8-c", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled c of Q₈, specified by its character on the named conjugacy classes. i maps to -1 and j to -1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#dimension", "subject": "representation:q-8-c", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#field", "subject": "representation:q-8-c", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#reality", "subject": "representation:q-8-c", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#indicator", "subject": "representation:q-8-c", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#conjugate", "subject": "representation:q-8-c", "key": "conjugate", "label": "Complex conjugate irrep", "value": "c", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#representations-1", "subject": "representation:q-8-c", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#physics-1", "subject": "representation:q-8-c", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#conventions-1", "subject": "representation:q-8-c", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-c#relation-1", "subject": "representation:q-8-c", "key": "relation-1", "label": "Representation domain", "value": "group:q-8", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep c is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#definition", "subject": "representation:q-8-2", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 2 of Q₈, specified by its character on the named conjugacy classes. Faithful quaternionic-type doublet; its character is real-valued", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#dimension", "subject": "representation:q-8-2", "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#field", "subject": "representation:q-8-2", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#reality", "subject": "representation:q-8-2", "key": "reality", "label": "Frobenius–Schur type", "value": "Quaternionic", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#indicator", "subject": "representation:q-8-2", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "-1", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#conjugate", "subject": "representation:q-8-2", "key": "conjugate", "label": "Complex conjugate irrep", "value": "2", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#representations-1", "subject": "representation:q-8-2", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j).", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#physics-1", "subject": "representation:q-8-2", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#conventions-1", "subject": "representation:q-8-2", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; quaternionic describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:q-8-2#relation-1", "subject": "representation:q-8-2", "key": "relation-1", "label": "Representation domain", "value": "group:q-8", "evidence": [ { "source": "groupnames-q-8", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of Q₈; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#definition", "subject": "representation:a-4-1", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1 of A₄, specified by its character on the named conjugacy classes. Trivial representation", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#dimension", "subject": "representation:a-4-1", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#field", "subject": "representation:a-4-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#reality", "subject": "representation:a-4-1", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#indicator", "subject": "representation:a-4-1", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#conjugate", "subject": "representation:a-4-1", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#representations-1", "subject": "representation:a-4-1", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123).", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#physics-1", "subject": "representation:a-4-1", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#conventions-1", "subject": "representation:a-4-1", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1#relation-1", "subject": "representation:a-4-1", "key": "relation-1", "label": "Representation domain", "value": "group:a-4", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#definition", "subject": "representation:a-4-1p", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1′ of A₄, specified by its character on the named conjugacy classes. Nontrivial quotient character with value omega on (123)", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#dimension", "subject": "representation:a-4-1p", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#field", "subject": "representation:a-4-1p", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#reality", "subject": "representation:a-4-1p", "key": "reality", "label": "Frobenius–Schur type", "value": "Complex", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#indicator", "subject": "representation:a-4-1p", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "0", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#conjugate", "subject": "representation:a-4-1p", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1″", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#representations-1", "subject": "representation:a-4-1p", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123).", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#physics-1", "subject": "representation:a-4-1p", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#conventions-1", "subject": "representation:a-4-1p", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; complex describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#relation-1", "subject": "representation:a-4-1p", "key": "relation-1", "label": "Representation domain", "value": "group:a-4", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1p#relation-2", "subject": "representation:a-4-1p", "key": "relation-2", "label": "Complex conjugate representation", "value": "representation:a-4-1pp", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "conjugate", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#definition", "subject": "representation:a-4-1pp", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1″ of A₄, specified by its character on the named conjugacy classes. Complex conjugate of 1′", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#dimension", "subject": "representation:a-4-1pp", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#field", "subject": "representation:a-4-1pp", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#reality", "subject": "representation:a-4-1pp", "key": "reality", "label": "Frobenius–Schur type", "value": "Complex", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#indicator", "subject": "representation:a-4-1pp", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "0", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#conjugate", "subject": "representation:a-4-1pp", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1′", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#representations-1", "subject": "representation:a-4-1pp", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123).", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#physics-1", "subject": "representation:a-4-1pp", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#conventions-1", "subject": "representation:a-4-1pp", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; complex describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#relation-1", "subject": "representation:a-4-1pp", "key": "relation-1", "label": "Representation domain", "value": "group:a-4", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-1pp#relation-2", "subject": "representation:a-4-1pp", "key": "relation-2", "label": "Complex conjugate representation", "value": "representation:a-4-1p", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 1″ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "conjugate", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#definition", "subject": "representation:a-4-3", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 3 of A₄, specified by its character on the named conjugacy classes. Faithful real-type rotational triplet", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#dimension", "subject": "representation:a-4-3", "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#field", "subject": "representation:a-4-3", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#reality", "subject": "representation:a-4-3", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#indicator", "subject": "representation:a-4-3", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#conjugate", "subject": "representation:a-4-3", "key": "conjugate", "label": "Complex conjugate irrep", "value": "3", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#representations-1", "subject": "representation:a-4-3", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are identity, double transpositions, the class of (123), and the class of (132). The label 1′ takes omega on (123).", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#physics-1", "subject": "representation:a-4-3", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#conventions-1", "subject": "representation:a-4-3", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-4-3#relation-1", "subject": "representation:a-4-3", "key": "relation-1", "label": "Representation domain", "value": "group:a-4", "evidence": [ { "source": "a4", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#definition", "subject": "representation:s-4-1", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1 of S₄, specified by its character on the named conjugacy classes. Trivial representation", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#dimension", "subject": "representation:s-4-1", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#field", "subject": "representation:s-4-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#reality", "subject": "representation:s-4-1", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#indicator", "subject": "representation:s-4-1", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#conjugate", "subject": "representation:s-4-1", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#representations-1", "subject": "representation:s-4-1", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#physics-1", "subject": "representation:s-4-1", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#conventions-1", "subject": "representation:s-4-1", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-1#relation-1", "subject": "representation:s-4-1", "key": "relation-1", "label": "Representation domain", "value": "group:s-4", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#definition", "subject": "representation:s-4-sgn", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled sgn of S₄, specified by its character on the named conjugacy classes. Sign of a permutation", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#dimension", "subject": "representation:s-4-sgn", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#field", "subject": "representation:s-4-sgn", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#reality", "subject": "representation:s-4-sgn", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#indicator", "subject": "representation:s-4-sgn", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#conjugate", "subject": "representation:s-4-sgn", "key": "conjugate", "label": "Complex conjugate irrep", "value": "sgn", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#representations-1", "subject": "representation:s-4-sgn", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#physics-1", "subject": "representation:s-4-sgn", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#conventions-1", "subject": "representation:s-4-sgn", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-sgn#relation-1", "subject": "representation:s-4-sgn", "key": "relation-1", "label": "Representation domain", "value": "group:s-4", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#definition", "subject": "representation:s-4-2", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 2 of S₄, specified by its character on the named conjugacy classes. Standard S3 doublet pulled back through S4 → S3", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#dimension", "subject": "representation:s-4-2", "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#field", "subject": "representation:s-4-2", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#reality", "subject": "representation:s-4-2", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#indicator", "subject": "representation:s-4-2", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#conjugate", "subject": "representation:s-4-2", "key": "conjugate", "label": "Complex conjugate irrep", "value": "2", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#representations-1", "subject": "representation:s-4-2", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#physics-1", "subject": "representation:s-4-2", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#conventions-1", "subject": "representation:s-4-2", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-2#relation-1", "subject": "representation:s-4-2", "key": "relation-1", "label": "Representation domain", "value": "group:s-4", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 2 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#definition", "subject": "representation:s-4-3", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 3 of S₄, specified by its character on the named conjugacy classes. Standard triplet; the determinant is sgn", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#dimension", "subject": "representation:s-4-3", "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#field", "subject": "representation:s-4-3", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#reality", "subject": "representation:s-4-3", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#indicator", "subject": "representation:s-4-3", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#conjugate", "subject": "representation:s-4-3", "key": "conjugate", "label": "Complex conjugate irrep", "value": "3", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#representations-1", "subject": "representation:s-4-3", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#physics-1", "subject": "representation:s-4-3", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#conventions-1", "subject": "representation:s-4-3", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3#relation-1", "subject": "representation:s-4-3", "key": "relation-1", "label": "Representation domain", "value": "group:s-4", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#definition", "subject": "representation:s-4-3p", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 3′ of S₄, specified by its character on the named conjugacy classes. Sign twist of the standard triplet; the cubic-rotation representation", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#dimension", "subject": "representation:s-4-3p", "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#field", "subject": "representation:s-4-3p", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#reality", "subject": "representation:s-4-3p", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#indicator", "subject": "representation:s-4-3p", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#conjugate", "subject": "representation:s-4-3p", "key": "conjugate", "label": "Complex conjugate irrep", "value": "3′", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#representations-1", "subject": "representation:s-4-3p", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#physics-1", "subject": "representation:s-4-3p", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#conventions-1", "subject": "representation:s-4-3p", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:s-4-3p#relation-1", "subject": "representation:s-4-3p", "key": "relation-1", "label": "Representation domain", "value": "group:s-4", "evidence": [ { "source": "groupnames-s-4", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of S₄; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#definition", "subject": "representation:a-5-1", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 1 of A₅, specified by its character on the named conjugacy classes. Trivial representation", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#dimension", "subject": "representation:a-5-1", "key": "dimension", "label": "Complex dimension", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#field", "subject": "representation:a-5-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#reality", "subject": "representation:a-5-1", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#indicator", "subject": "representation:a-5-1", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#conjugate", "subject": "representation:a-5-1", "key": "conjugate", "label": "Complex conjugate irrep", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#representations-1", "subject": "representation:a-5-1", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#physics-1", "subject": "representation:a-5-1", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#conventions-1", "subject": "representation:a-5-1", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-1#relation-1", "subject": "representation:a-5-1", "key": "relation-1", "label": "Representation domain", "value": "group:a-5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#definition", "subject": "representation:a-5-3", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 3 of A₅, specified by its character on the named conjugacy classes. Real-type triplet with phi on the chosen five-cycle c", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#dimension", "subject": "representation:a-5-3", "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#field", "subject": "representation:a-5-3", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#reality", "subject": "representation:a-5-3", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#indicator", "subject": "representation:a-5-3", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#conjugate", "subject": "representation:a-5-3", "key": "conjugate", "label": "Complex conjugate irrep", "value": "3", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#representations-1", "subject": "representation:a-5-3", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#physics-1", "subject": "representation:a-5-3", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#conventions-1", "subject": "representation:a-5-3", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3#relation-1", "subject": "representation:a-5-3", "key": "relation-1", "label": "Representation domain", "value": "group:a-5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#definition", "subject": "representation:a-5-3p", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 3′ of A₅, specified by its character on the named conjugacy classes. Inequivalent real-type triplet obtained by exchanging five-cycle classes", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#dimension", "subject": "representation:a-5-3p", "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#field", "subject": "representation:a-5-3p", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#reality", "subject": "representation:a-5-3p", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#indicator", "subject": "representation:a-5-3p", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#conjugate", "subject": "representation:a-5-3p", "key": "conjugate", "label": "Complex conjugate irrep", "value": "3′", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#representations-1", "subject": "representation:a-5-3p", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#physics-1", "subject": "representation:a-5-3p", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#conventions-1", "subject": "representation:a-5-3p", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-3p#relation-1", "subject": "representation:a-5-3p", "key": "relation-1", "label": "Representation domain", "value": "group:a-5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 3′ is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#definition", "subject": "representation:a-5-4", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 4 of A₅, specified by its character on the named conjugacy classes. Five-letter permutation representation minus the invariant line", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#dimension", "subject": "representation:a-5-4", "key": "dimension", "label": "Complex dimension", "value": "4", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#field", "subject": "representation:a-5-4", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#reality", "subject": "representation:a-5-4", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#indicator", "subject": "representation:a-5-4", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#conjugate", "subject": "representation:a-5-4", "key": "conjugate", "label": "Complex conjugate irrep", "value": "4", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#representations-1", "subject": "representation:a-5-4", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#physics-1", "subject": "representation:a-5-4", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#conventions-1", "subject": "representation:a-5-4", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-4#relation-1", "subject": "representation:a-5-4", "key": "relation-1", "label": "Representation domain", "value": "group:a-5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 4 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#definition", "subject": "representation:a-5-5", "key": "definition", "label": "Definition", "value": "The ordinary complex irreducible representation labeled 5 of A₅, specified by its character on the named conjugacy classes. Real-type five-dimensional irrep", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#dimension", "subject": "representation:a-5-5", "key": "dimension", "label": "Complex dimension", "value": "5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#field", "subject": "representation:a-5-5", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#reality", "subject": "representation:a-5-5", "key": "reality", "label": "Frobenius–Schur type", "value": "Real", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#indicator", "subject": "representation:a-5-5", "key": "indicator", "label": "Frobenius–Schur indicator", "value": "1", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#conjugate", "subject": "representation:a-5-5", "key": "conjugate", "label": "Complex conjugate irrep", "value": "5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#representations-1", "subject": "representation:a-5-5", "key": "representations-1", "label": "Character and identity", "value": "This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns are 1,(12)(34),(123),c,c², with c=(12345). The triplet 3 has value phi on c; 3′ exchanges the two five-cycle values. Squaring exchanges the five-cycle classes.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#physics-1", "subject": "representation:a-5-5", "key": "physics-1", "label": "Tensor operations and physical use", "value": "Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.", "evidence": [ { "source": "finite-etingof", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators" }, { "source": "gap", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#conventions-1", "subject": "representation:a-5-5", "key": "conventions-1", "label": "Conventions and realization", "value": "The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:a-5-5#relation-1", "subject": "representation:a-5-5", "key": "relation-1", "label": "Representation domain", "value": "group:a-5", "evidence": [ { "source": "groupnames-a-5", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-etingof", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "gap", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "indicators", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." }, { "source": "finite-gap-power-maps", "locator": "Ordinary character table of A₅; the irrep 5 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#definition", "subject": "representation:su-2-weight-1", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of SU(2) with Dynkin labels (1) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#dimension", "subject": "representation:su-2-weight-1", "key": "dimension", "label": "Complex dimension", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#field", "subject": "representation:su-2-weight-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#casimir", "subject": "representation:su-2-weight-1", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "3/4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#index", "subject": "representation:su-2-weight-1", "key": "index", "label": "Quadratic index T(R)", "value": "1/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#reality", "subject": "representation:su-2-weight-1", "key": "reality", "label": "Reality type", "value": "Quaternionic", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#conjugate", "subject": "representation:su-2-weight-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#adjoint-descent", "subject": "representation:su-2-weight-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#central-kernel-order", "subject": "representation:su-2-weight-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#representations-1", "subject": "representation:su-2-weight-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#global-form-1", "subject": "representation:su-2-weight-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#physics-1", "subject": "representation:su-2-weight-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-1#relation-1", "subject": "representation:su-2-weight-1", "key": "relation-1", "label": "Representation domain", "value": "group:su-2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#definition", "subject": "representation:su-2-weight-2", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of SU(2) with Dynkin labels (2) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#dimension", "subject": "representation:su-2-weight-2", "key": "dimension", "label": "Complex dimension", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#field", "subject": "representation:su-2-weight-2", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#casimir", "subject": "representation:su-2-weight-2", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#index", "subject": "representation:su-2-weight-2", "key": "index", "label": "Quadratic index T(R)", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#reality", "subject": "representation:su-2-weight-2", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#conjugate", "subject": "representation:su-2-weight-2", "key": "conjugate", "label": "Conjugate highest weight", "value": "(2)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#adjoint-descent", "subject": "representation:su-2-weight-2", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#central-kernel-order", "subject": "representation:su-2-weight-2", "key": "central-kernel-order", "label": "Order of central kernel", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#representations-1", "subject": "representation:su-2-weight-2", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#global-form-1", "subject": "representation:su-2-weight-2", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#physics-1", "subject": "representation:su-2-weight-2", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-2-weight-2#relation-1", "subject": "representation:su-2-weight-2", "key": "relation-1", "label": "Representation domain", "value": "group:su-2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-0#field", "subject": "representation:su-3-weight-1-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-0#casimir", "subject": "representation:su-3-weight-1-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "4/3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-0#index", "subject": "representation:su-3-weight-1-0", "key": "index", "label": "Quadratic index T(R)", "value": "1/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-0#reality", "subject": "representation:su-3-weight-1-0", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-0#conjugate", "subject": "representation:su-3-weight-1-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-0#adjoint-descent", "subject": "representation:su-3-weight-1-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-0#central-kernel-order", "subject": "representation:su-3-weight-1-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-0-1#reality", "subject": "representation:su-3-weight-0-1", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#dimension", "subject": "representation:su-3-weight-1-1", "key": "dimension", "label": "Complex dimension", "value": "8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#casimir", "subject": "representation:su-3-weight-1-1", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#index", "subject": "representation:su-3-weight-1-1", "key": "index", "label": "Quadratic index T(R)", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#reality", "subject": "representation:su-3-weight-1-1", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#conjugate", "subject": "representation:su-3-weight-1-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#adjoint-descent", "subject": "representation:su-3-weight-1-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#central-kernel-order", "subject": "representation:su-3-weight-1-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#representations-1", "subject": "representation:su-3-weight-1-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#global-form-1", "subject": "representation:su-3-weight-1-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#physics-1", "subject": "representation:su-3-weight-1-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-1-1#relation-1", "subject": "representation:su-3-weight-1-1", "key": "relation-1", "label": "Representation domain", "value": "group:su-3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#definition", "subject": "representation:su-3-weight-2-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of SU(3) with Dynkin labels (2,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#dimension", "subject": "representation:su-3-weight-2-0", "key": "dimension", "label": "Complex dimension", "value": "6", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#field", "subject": "representation:su-3-weight-2-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#casimir", "subject": "representation:su-3-weight-2-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "10/3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#index", "subject": "representation:su-3-weight-2-0", "key": "index", "label": "Quadratic index T(R)", "value": "5/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#reality", "subject": "representation:su-3-weight-2-0", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#conjugate", "subject": "representation:su-3-weight-2-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,2)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#adjoint-descent", "subject": "representation:su-3-weight-2-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#central-kernel-order", "subject": "representation:su-3-weight-2-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-0#representations-1", "subject": "representation:su-3-weight-2-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-3-0#index", "subject": "representation:su-3-weight-3-0", "key": "index", "label": "Quadratic index T(R)", "value": "15/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-3-0#reality", "subject": "representation:su-3-weight-3-0", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-3-0#conjugate", "subject": "representation:su-3-weight-3-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,3)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-3-0#adjoint-descent", "subject": "representation:su-3-weight-3-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-3-0#central-kernel-order", "subject": "representation:su-3-weight-3-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-3-0#representations-1", "subject": "representation:su-3-weight-3-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#casimir", "subject": "representation:su-3-weight-2-2", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#index", "subject": "representation:su-3-weight-2-2", "key": "index", "label": "Quadratic index T(R)", "value": "27", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#reality", "subject": "representation:su-3-weight-2-2", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#conjugate", "subject": "representation:su-3-weight-2-2", "key": "conjugate", "label": "Conjugate highest weight", "value": "(2,2)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#adjoint-descent", "subject": "representation:su-3-weight-2-2", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#central-kernel-order", "subject": "representation:su-3-weight-2-2", "key": "central-kernel-order", "label": "Order of central kernel", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-3", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#representations-1", "subject": "representation:su-3-weight-2-2", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#global-form-1", "subject": "representation:su-3-weight-2-2", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#physics-1", "subject": "representation:su-3-weight-2-2", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-3-weight-2-2#relation-1", "subject": "representation:su-3-weight-2-2", "key": "relation-1", "label": "Representation domain", "value": "group:su-3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#definition", "subject": "representation:su-4-weight-1-0-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of SU(4) with Dynkin labels (1,0,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#dimension", "subject": "representation:su-4-weight-1-0-0", "key": "dimension", "label": "Complex dimension", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:su-4", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#field", "subject": "representation:su-4-weight-1-0-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#casimir", "subject": "representation:su-4-weight-1-0-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "15/8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-4", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#index", "subject": "representation:su-4-weight-1-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "1/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-4", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#reality", "subject": "representation:su-4-weight-1-0-0", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#conjugate", "subject": "representation:su-4-weight-1-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#adjoint-descent", "subject": "representation:su-4-weight-1-0-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-4", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#central-kernel-order", "subject": "representation:su-4-weight-1-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-4", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-0#representations-1", "subject": "representation:su-4-weight-1-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-0-1-0#conjugate", "subject": "representation:su-4-weight-0-1-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-4", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-1#reality", "subject": "representation:su-4-weight-1-0-1", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-1#conjugate", "subject": "representation:su-4-weight-1-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-1#adjoint-descent", "subject": "representation:su-4-weight-1-0-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-4", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-1#central-kernel-order", "subject": "representation:su-4-weight-1-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-4", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-1#representations-1", "subject": "representation:su-4-weight-1-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-1#global-form-1", "subject": "representation:su-4-weight-1-0-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-1#physics-1", "subject": "representation:su-4-weight-1-0-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-4-weight-1-0-1#relation-1", "subject": "representation:su-4-weight-1-0-1", "key": "relation-1", "label": "Representation domain", "value": "group:su-4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#definition", "subject": "representation:su-5-weight-1-0-0-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of SU(5) with Dynkin labels (1,0,0,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#dimension", "subject": "representation:su-5-weight-1-0-0-0", "key": "dimension", "label": "Complex dimension", "value": "5", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#field", "subject": "representation:su-5-weight-1-0-0-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#casimir", "subject": "representation:su-5-weight-1-0-0-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "12/5", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#index", "subject": "representation:su-5-weight-1-0-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "1/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#reality", "subject": "representation:su-5-weight-1-0-0-0", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#conjugate", "subject": "representation:su-5-weight-1-0-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#adjoint-descent", "subject": "representation:su-5-weight-1-0-0-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#central-kernel-order", "subject": "representation:su-5-weight-1-0-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#representations-1", "subject": "representation:su-5-weight-1-0-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#global-form-1", "subject": "representation:su-5-weight-1-0-0-0", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-0#physics-1", "subject": "representation:su-5-weight-1-0-0-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-0-1-0-0#central-kernel-order", "subject": "representation:su-5-weight-0-1-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-0-1-0-0#representations-1", "subject": "representation:su-5-weight-0-1-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. 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The test for descent to its adjoint global form gives no. 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A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-1#conjugate", "subject": "representation:su-5-weight-1-0-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-1#central-kernel-order", "subject": "representation:su-5-weight-1-0-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "5", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:su-5", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-1#representations-1", "subject": "representation:su-5-weight-1-0-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain. Fundamental weight 1 is the defining representation of SU(r+1). Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-1#global-form-1", "subject": "representation:su-5-weight-1-0-0-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:su-5-weight-1-0-0-1#physics-1", "subject": "representation:su-5-weight-1-0-0-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#definition", "subject": "representation:sp-2-weight-1-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of Sp(2) = USp(4) with Dynkin labels (1,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#dimension", "subject": "representation:sp-2-weight-1-0", "key": "dimension", "label": "Complex dimension", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#field", "subject": "representation:sp-2-weight-1-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#casimir", "subject": "representation:sp-2-weight-1-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "5/4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#index", "subject": "representation:sp-2-weight-1-0", "key": "index", "label": "Quadratic index T(R)", "value": "1/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#reality", "subject": "representation:sp-2-weight-1-0", "key": "reality", "label": "Reality type", "value": "Quaternionic", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#conjugate", "subject": "representation:sp-2-weight-1-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#adjoint-descent", "subject": "representation:sp-2-weight-1-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#central-kernel-order", "subject": "representation:sp-2-weight-1-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#representations-1", "subject": "representation:sp-2-weight-1-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain; node r is long. Sp(r) denotes the compact quaternionic group. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#global-form-1", "subject": "representation:sp-2-weight-1-0", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. 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A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-1-0#relation-1", "subject": "representation:sp-2-weight-1-0", "key": "relation-1", "label": "Representation domain", "value": "group:sp-2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#definition", "subject": "representation:sp-2-weight-0-1", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of Sp(2) = USp(4) with Dynkin labels (0,1) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#dimension", "subject": "representation:sp-2-weight-0-1", "key": "dimension", "label": "Complex dimension", "value": "5", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#field", "subject": "representation:sp-2-weight-0-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#casimir", "subject": "representation:sp-2-weight-0-1", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#index", "subject": "representation:sp-2-weight-0-1", "key": "index", "label": "Quadratic index T(R)", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#reality", "subject": "representation:sp-2-weight-0-1", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#conjugate", "subject": "representation:sp-2-weight-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#adjoint-descent", "subject": "representation:sp-2-weight-0-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#central-kernel-order", "subject": "representation:sp-2-weight-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:sp-2", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-c", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#representations-1", "subject": "representation:sp-2-weight-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "Nodes 1,…,r run along the chain; node r is long. Sp(r) denotes the compact quaternionic group. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#global-form-1", "subject": "representation:sp-2-weight-0-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#physics-1", "subject": "representation:sp-2-weight-0-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:sp-2-weight-0-1#relation-1", "subject": "representation:sp-2-weight-0-1", "key": "relation-1", "label": "Representation domain", "value": "group:sp-2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-c", "locator": "AmbientSpace.simple_root(): final node is long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#definition", "subject": "representation:spin-8-weight-1-0-0-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of Spin(8) with Dynkin labels (1,0,0,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#dimension", "subject": "representation:spin-8-weight-1-0-0-0", "key": "dimension", "label": "Complex dimension", "value": "8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#field", "subject": "representation:spin-8-weight-1-0-0-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#casimir", "subject": "representation:spin-8-weight-1-0-0-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "7/2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#index", "subject": "representation:spin-8-weight-1-0-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#reality", "subject": "representation:spin-8-weight-1-0-0-0", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#conjugate", "subject": "representation:spin-8-weight-1-0-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#central-kernel-order", "subject": "representation:spin-8-weight-1-0-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#representations-1", "subject": "representation:spin-8-weight-1-0-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-1-0-0-0#physics-1", "subject": "representation:spin-8-weight-1-0-0-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#dimension", "subject": "representation:spin-8-weight-0-0-1-0", "key": "dimension", "label": "Complex dimension", "value": "8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#field", "subject": "representation:spin-8-weight-0-0-1-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#index", "subject": "representation:spin-8-weight-0-0-1-0", "key": "index", "label": "Quadratic index T(R)", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#reality", "subject": "representation:spin-8-weight-0-0-1-0", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#conjugate", "subject": "representation:spin-8-weight-0-0-1-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,1,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#adjoint-descent", "subject": "representation:spin-8-weight-0-0-1-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#central-kernel-order", "subject": "representation:spin-8-weight-0-0-1-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#representations-1", "subject": "representation:spin-8-weight-0-0-1-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-1-0#physics-1", "subject": "representation:spin-8-weight-0-0-1-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-0-1#reality", "subject": "representation:spin-8-weight-0-0-0-1", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-0-1#conjugate", "subject": "representation:spin-8-weight-0-0-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-0-1#adjoint-descent", "subject": "representation:spin-8-weight-0-0-0-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-0-1#central-kernel-order", "subject": "representation:spin-8-weight-0-0-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-8", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-0-1#representations-1", "subject": "representation:spin-8-weight-0-0-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-0-1#global-form-1", "subject": "representation:spin-8-weight-0-0-0-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-8-weight-0-0-0-1#physics-1", "subject": "representation:spin-8-weight-0-0-0-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-1-0-0-0-0#index", "subject": "representation:spin-10-weight-1-0-0-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-1-0-0-0-0#reality", "subject": "representation:spin-10-weight-1-0-0-0-0", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-1-0-0-0-0#conjugate", "subject": "representation:spin-10-weight-1-0-0-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-1-0-0-0-0#adjoint-descent", "subject": "representation:spin-10-weight-1-0-0-0-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-1-0-0-0-0#central-kernel-order", "subject": "representation:spin-10-weight-1-0-0-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-1-0-0-0-0#representations-1", "subject": "representation:spin-10-weight-1-0-0-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-1-0-0-0-0#physics-1", "subject": "representation:spin-10-weight-1-0-0-0-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-1-0-0-0-0#relation-1", "subject": "representation:spin-10-weight-1-0-0-0-0", "key": "relation-1", "label": "Representation domain", "value": "group:spin-10", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#definition", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of Spin(10) with Dynkin labels (0,0,0,0,1) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#dimension", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "dimension", "label": "Complex dimension", "value": "16", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#field", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#casimir", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "45/8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#index", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "index", "label": "Quadratic index T(R)", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#reality", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#conjugate", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,1,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#adjoint-descent", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#central-kernel-order", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-0-1#representations-1", "subject": "representation:spin-10-weight-0-0-0-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-1-0#reality", "subject": "representation:spin-10-weight-0-0-0-1-0", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-1-0#central-kernel-order", "subject": "representation:spin-10-weight-0-0-0-1-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-0-0-1-0#representations-1", "subject": "representation:spin-10-weight-0-0-0-1-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. 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documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#index", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "8", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#reality", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#conjugate", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#adjoint-descent", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#central-kernel-order", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:spin-10", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#representations-1", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#global-form-1", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#physics-1", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:spin-10-weight-0-1-0-0-0#relation-1", "subject": "representation:spin-10-weight-0-1-0-0-0", "key": "relation-1", "label": "Representation domain", "value": "group:spin-10", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#definition", "subject": "representation:g-2-compact-weight-1-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of G2 (compact) with Dynkin labels (1,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#dimension", "subject": "representation:g-2-compact-weight-1-0", "key": "dimension", "label": "Complex dimension", "value": "7", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#field", "subject": "representation:g-2-compact-weight-1-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#casimir", "subject": "representation:g-2-compact-weight-1-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#index", "subject": "representation:g-2-compact-weight-1-0", "key": "index", "label": "Quadratic index T(R)", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#reality", "subject": "representation:g-2-compact-weight-1-0", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#conjugate", "subject": "representation:g-2-compact-weight-1-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#adjoint-descent", "subject": "representation:g-2-compact-weight-1-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#central-kernel-order", "subject": "representation:g-2-compact-weight-1-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#representations-1", "subject": "representation:g-2-compact-weight-1-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "Node 1 is short and node 2 long. Weights 1 and 2 have dimensions 7 and 14, respectively. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#global-form-1", "subject": "representation:g-2-compact-weight-1-0", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-1-0#physics-1", "subject": "representation:g-2-compact-weight-1-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#definition", "subject": "representation:g-2-compact-weight-0-1", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of G2 (compact) with Dynkin labels (0,1) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#dimension", "subject": "representation:g-2-compact-weight-0-1", "key": "dimension", "label": "Complex dimension", "value": "14", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#field", "subject": "representation:g-2-compact-weight-0-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#casimir", "subject": "representation:g-2-compact-weight-0-1", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#index", "subject": "representation:g-2-compact-weight-0-1", "key": "index", "label": "Quadratic index T(R)", "value": "4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#reality", "subject": "representation:g-2-compact-weight-0-1", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#conjugate", "subject": "representation:g-2-compact-weight-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#adjoint-descent", "subject": "representation:g-2-compact-weight-0-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#central-kernel-order", "subject": "representation:g-2-compact-weight-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:g-2-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-g", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#representations-1", "subject": "representation:g-2-compact-weight-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "Node 1 is short and node 2 long. Weights 1 and 2 have dimensions 7 and 14, respectively. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#global-form-1", "subject": "representation:g-2-compact-weight-0-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-g", "locator": "AmbientSpace.simple_root(), fundamental_weights(): node 1 short and node 2 long." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:g-2-compact-weight-0-1#physics-1", "subject": "representation:g-2-compact-weight-0-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#dimension", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "dimension", "label": "Complex dimension", "value": "26", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#field", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#casimir", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "6", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#index", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "index", "label": "Quadratic index T(R)", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#reality", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#conjugate", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#adjoint-descent", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#central-kernel-order", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#representations-1", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain is 1–2–3–4; nodes 1 and 2 are long, nodes 3 and 4 short. Weight 4 has dimension 26. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#global-form-1", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#physics-1", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-0-0-0-1#relation-1", "subject": "representation:f-4-compact-weight-0-0-0-1", "key": "relation-1", "label": "Representation domain", "value": "group:f-4-compact", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#definition", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of F4 (compact) with Dynkin labels (1,0,0,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#dimension", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "dimension", "label": "Complex dimension", "value": "52", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#field", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#casimir", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "9", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#index", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "9", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#reality", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#conjugate", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#adjoint-descent", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#central-kernel-order", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:f-4-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-f", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#representations-1", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The chain is 1–2–3–4; nodes 1 and 2 are long, nodes 3 and 4 short. Weight 4 has dimension 26. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#global-form-1", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#physics-1", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:f-4-compact-weight-1-0-0-0#relation-1", "subject": "representation:f-4-compact-weight-1-0-0-0", "key": "relation-1", "label": "Representation domain", "value": "group:f-4-compact", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-f", "locator": "AmbientSpace.simple_root(): nodes 1,2 long and nodes 3,4 short." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#definition", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of E6 (compact simply connected) with Dynkin labels (1,0,0,0,0,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#dimension", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "dimension", "label": "Complex dimension", "value": "27", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#field", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#casimir", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "26/3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#index", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#reality", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "reality", "label": "Reality type", "value": "Complex", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#conjugate", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#adjoint-descent", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#central-kernel-order", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-1-0-0-0-0-0#representations-1", "subject": "representation:e-6-compact-sc-weight-1-0-0-0-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. 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The test for descent to its adjoint global form gives no. 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A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#definition", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of E6 (compact simply connected) with Dynkin labels (0,1,0,0,0,0) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#dimension", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "dimension", "label": "Complex dimension", "value": "78", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#field", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#casimir", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "12", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#index", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "12", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#reality", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#conjugate", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,1,0,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#adjoint-descent", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#central-kernel-order", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "3", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-6-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#representations-1", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#global-form-1", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-6-compact-sc-weight-0-1-0-0-0-0#physics-1", "subject": "representation:e-6-compact-sc-weight-0-1-0-0-0-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. 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documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#definition", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of E7 (compact simply connected) with Dynkin labels (0,0,0,0,0,0,1) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#dimension", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "dimension", "label": "Complex dimension", "value": "56", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#field", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#casimir", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "57/4", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#index", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "index", "label": "Quadratic index T(R)", "value": "6", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#reality", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "reality", "label": "Reality type", "value": "Quaternionic", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#conjugate", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#adjoint-descent", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "No", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#central-kernel-order", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#representations-1", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#global-form-1", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. 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A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1#relation-1", "subject": "representation:e-7-compact-sc-weight-0-0-0-0-0-0-1", "key": "relation-1", "label": "Representation domain", "value": "group:e-7-compact-sc", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; 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characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#casimir", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "18", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#index", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "index", "label": "Quadratic index T(R)", "value": "18", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#reality", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#conjugate", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "conjugate", "label": "Conjugate highest weight", "value": "(1,0,0,0,0,0,0)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#adjoint-descent", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#central-kernel-order", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "central-kernel-order", "label": "Order of central kernel", "value": "2", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-7-compact-sc", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#representations-1", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "representations-1", "label": "Highest weight and conventions", "value": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#global-form-1", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#physics-1", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0#relation-1", "subject": "representation:e-7-compact-sc-weight-1-0-0-0-0-0-0", "key": "relation-1", "label": "Representation domain", "value": "group:e-7-compact-sc", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#definition", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "definition", "label": "Definition", "value": "The finite-dimensional continuous complex irreducible representation of E8 (compact simply connected) with Dynkin labels (0,0,0,0,0,0,0,1) in the displayed Bourbaki/Sage node order.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#dimension", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "dimension", "label": "Complex dimension", "value": "248", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Exact Weyl dimension product over positive coroots", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#field", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "field", "label": "Coefficient field", "value": "ℂ; characteristic zero", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#casimir", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "casimir", "label": "Quadratic Casimir C₂", "value": "30", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "(lambda,lambda+2rho)/2 with squared long-root length two", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#index", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "index", "label": "Quadratic index T(R)", "value": "30", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "T(R) dim(g) = C2(R) dim(R)", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#reality", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "reality", "label": "Reality type", "value": "Real", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#conjugate", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "conjugate", "label": "Conjugate highest weight", "value": "(0,0,0,0,0,0,0,1)", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#adjoint-descent", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "adjoint-descent", "label": "Descends to the adjoint global form", "value": "Yes", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Highest weight lies in the root lattice exactly when the full center acts trivially", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#central-kernel-order", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "central-kernel-order", "label": "Order of central kernel", "value": "1", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "derivation": { "method": "Order of the center divided by the order of this central character", "inputs": [ "group:e-8-compact", "source:workbench-etingof-lie", "source:workbench-sage-roots", "source:workbench-sage-e", "source:sage", "source:conrad-centers", "source:workbench-adams-indicator" ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#representations-1", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "representations-1", "label": "Highest weight and conventions", "value": "The horizontal chain is 1–3–4–5–6 (continuing to 7 and 8 when present); node 2 attaches to node 4. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#global-form-1", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "global-form-1", "label": "Global form and center", "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives yes. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#physics-1", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "physics-1", "label": "Use in a calculation", "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1#relation-1", "subject": "representation:e-8-compact-weight-0-0-0-0-0-0-0-1", "key": "relation-1", "label": "Representation domain", "value": "group:e-8-compact", "evidence": [ { "source": "workbench-etingof-lie", "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations." }, { "source": "workbench-sage-roots", "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08." }, { "source": "workbench-sage-e", "locator": "AmbientSpace and CartanType.dynkin_diagram(): Bourbaki nodes, with node 2 attached to node 4." }, { "source": "sage", "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL." }, { "source": "conrad-centers", "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated." }, { "source": "workbench-adams-indicator", "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep." } ], "assumptions": [], "relationship": "representation-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#definition", "subject": "algebra:standard-model-r", "key": "definition", "label": "Definition", "value": "The direct sum of the compact real algebras su(3), su(2), and u(1). Its real dimension is 8+3+1=12, its rank is 2+1+1=4, and its one-dimensional center makes it reductive but not semisimple.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#dimension", "subject": "algebra:standard-model-r", "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#rank", "subject": "algebra:standard-model-r", "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#center", "subject": "algebra:standard-model-r", "key": "center", "label": "Lie-algebra center", "value": "u(1), dimension one", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#simple", "subject": "algebra:standard-model-r", "key": "simple", "label": "Simple", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#global-forms-1", "subject": "algebra:standard-model-r", "key": "global-forms-1", "label": "Global groups and charge normalization", "value": "The Lie algebra alone does not fix a connected global gauge group or the allowed line operators. The four group records here use the compact product SU(3) × SU(2) × U(1), integer charge Q=6Y, and subgroups of its specified diagonal Z6.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#relation-1", "subject": "algebra:standard-model-r", "key": "relation-1", "label": "Color algebra", "value": "algebra:su-3-r", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-factor", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#relation-2", "subject": "algebra:standard-model-r", "key": "relation-2", "label": "Weak algebra", "value": "algebra:su-2-r", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-factor", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#relation-3", "subject": "algebra:standard-model-r", "key": "relation-3", "label": "Abelian algebra", "value": "algebra:u-1-r", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-factor", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#relation-4", "subject": "algebra:standard-model-r", "key": "relation-4", "label": "Compact global form with quotient order 1", "value": "group:standard-model-z1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#relation-5", "subject": "algebra:standard-model-r", "key": "relation-5", "label": "Compact global form with quotient order 2", "value": "group:standard-model-z2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#relation-6", "subject": "algebra:standard-model-r", "key": "relation-6", "label": "Compact global form with quotient order 3", "value": "group:standard-model-z3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "algebra:standard-model-r#relation-7", "subject": "algebra:standard-model-r", "key": "relation-7", "label": "Compact global form with quotient order 6", "value": "group:standard-model-z6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "integrates-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#definition", "subject": "group:standard-model-z1", "key": "definition", "label": "Definition", "value": "The compact product SU(3) × SU(2) × U(1), with U(1) period 2π and integer hypercharge Q=6Y. It is a covering group of the three nontrivial specified quotients, but it is not the universal cover.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#dimension", "subject": "group:standard-model-z1", "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#rank", "subject": "group:standard-model-z1", "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#compact", "subject": "group:standard-model-z1", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#connected", "subject": "group:standard-model-z1", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#pi1", "subject": "group:standard-model-z1", "key": "pi1", "label": "Fundamental group", "value": "Z", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#quotient-order", "subject": "group:standard-model-z1", "key": "quotient-order", "label": "Specified quotient order", "value": "1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#center", "subject": "group:standard-model-z1", "key": "center", "label": "Abstract center", "value": "U(1) × Z_6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "derivation": { "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.", "inputs": [ "source:conrad-pi1-lie", "Γd projects injectively to both the diagonal finite center and the U(1) factor." ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#representations-1", "subject": "group:standard-model-z1", "key": "representations-1", "label": "Matter and the embedded kernel", "value": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/1) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#representations-2", "subject": "group:standard-model-z1", "key": "representations-2", "label": "Matter and the embedded kernel", "value": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#topology-1", "subject": "group:standard-model-z1", "key": "topology-1", "label": "Topology and the universal cover", "value": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_1 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_1. The compact product used as the quotient domain already has fundamental group Z.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#conventions-1", "subject": "group:standard-model-z1", "key": "conventions-1", "label": "Hypercharge and SU(5)", "value": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#relation-1", "subject": "group:standard-model-z1", "key": "relation-1", "label": "The common gauge algebra", "value": "algebra:standard-model-r", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#relation-2", "subject": "group:standard-model-z1", "key": "relation-2", "label": "Diagonal quotient of order 2", "value": "group:standard-model-z2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#relation-3", "subject": "group:standard-model-z1", "key": "relation-3", "label": "Diagonal quotient of order 3", "value": "group:standard-model-z3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z1#relation-4", "subject": "group:standard-model-z1", "key": "relation-4", "label": "Diagonal quotient of order 6", "value": "group:standard-model-z6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-central-quotient", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#definition", "subject": "group:standard-model-z2", "key": "definition", "label": "Definition", "value": "The group (SU(3) × SU(2) × U(1))/Γ_2, where Γ_2=⟨γ^3⟩ and γ=(exp(2πi/3) I₃, −I₂, exp(2πi/6)). The U(1) action on a matter multiplet is z^Q with integer Q=6Y.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#dimension", "subject": "group:standard-model-z2", "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#rank", "subject": "group:standard-model-z2", "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#compact", "subject": "group:standard-model-z2", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#connected", "subject": "group:standard-model-z2", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#pi1", "subject": "group:standard-model-z2", "key": "pi1", "label": "Fundamental group", "value": "Z", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#quotient-order", "subject": "group:standard-model-z2", "key": "quotient-order", "label": "Specified quotient order", "value": "2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#center", "subject": "group:standard-model-z2", "key": "center", "label": "Abstract center", "value": "U(1) × Z_3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "derivation": { "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.", "inputs": [ "source:conrad-pi1-lie", "Γd projects injectively to both the diagonal finite center and the U(1) factor." ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#representations-1", "subject": "group:standard-model-z2", "key": "representations-1", "label": "Matter and the embedded kernel", "value": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/2) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#representations-2", "subject": "group:standard-model-z2", "key": "representations-2", "label": "Matter and the embedded kernel", "value": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#topology-1", "subject": "group:standard-model-z2", "key": "topology-1", "label": "Topology and the universal cover", "value": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_2 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_2. The compact product used as the quotient domain already has fundamental group Z.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#conventions-1", "subject": "group:standard-model-z2", "key": "conventions-1", "label": "Hypercharge and SU(5)", "value": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#relation-1", "subject": "group:standard-model-z2", "key": "relation-1", "label": "The common gauge algebra", "value": "algebra:standard-model-r", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z2#relation-2", "subject": "group:standard-model-z2", "key": "relation-2", "label": "Specified compact covering product", "value": "group:standard-model-z1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "central-quotient-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#definition", "subject": "group:standard-model-z3", "key": "definition", "label": "Definition", "value": "The group (SU(3) × SU(2) × U(1))/Γ_3, where Γ_3=⟨γ^2⟩ and γ=(exp(2πi/3) I₃, −I₂, exp(2πi/6)). The U(1) action on a matter multiplet is z^Q with integer Q=6Y.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#dimension", "subject": "group:standard-model-z3", "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#rank", "subject": "group:standard-model-z3", "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#compact", "subject": "group:standard-model-z3", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#connected", "subject": "group:standard-model-z3", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#pi1", "subject": "group:standard-model-z3", "key": "pi1", "label": "Fundamental group", "value": "Z", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#quotient-order", "subject": "group:standard-model-z3", "key": "quotient-order", "label": "Specified quotient order", "value": "3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#center", "subject": "group:standard-model-z3", "key": "center", "label": "Abstract center", "value": "U(1) × Z_2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "derivation": { "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.", "inputs": [ "source:conrad-pi1-lie", "Γd projects injectively to both the diagonal finite center and the U(1) factor." ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#representations-1", "subject": "group:standard-model-z3", "key": "representations-1", "label": "Matter and the embedded kernel", "value": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/3) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#representations-2", "subject": "group:standard-model-z3", "key": "representations-2", "label": "Matter and the embedded kernel", "value": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#topology-1", "subject": "group:standard-model-z3", "key": "topology-1", "label": "Topology and the universal cover", "value": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_3 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_3. The compact product used as the quotient domain already has fundamental group Z.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#conventions-1", "subject": "group:standard-model-z3", "key": "conventions-1", "label": "Hypercharge and SU(5)", "value": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#relation-1", "subject": "group:standard-model-z3", "key": "relation-1", "label": "The common gauge algebra", "value": "algebra:standard-model-r", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z3#relation-2", "subject": "group:standard-model-z3", "key": "relation-2", "label": "Specified compact covering product", "value": "group:standard-model-z1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "central-quotient-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#definition", "subject": "group:standard-model-z6", "key": "definition", "label": "Definition", "value": "The group (SU(3) × SU(2) × U(1))/Γ_6, where Γ_6=⟨γ^1⟩ and γ=(exp(2πi/3) I₃, −I₂, exp(2πi/6)). The U(1) action on a matter multiplet is z^Q with integer Q=6Y.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#dimension", "subject": "group:standard-model-z6", "key": "dimension", "label": "Real dimension", "value": "12", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#rank", "subject": "group:standard-model-z6", "key": "rank", "label": "Rank", "value": "4", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#compact", "subject": "group:standard-model-z6", "key": "compact", "label": "Compact", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#connected", "subject": "group:standard-model-z6", "key": "connected", "label": "Connected", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#pi1", "subject": "group:standard-model-z6", "key": "pi1", "label": "Fundamental group", "value": "Z", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#quotient-order", "subject": "group:standard-model-z6", "key": "quotient-order", "label": "Specified quotient order", "value": "6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#center", "subject": "group:standard-model-z6", "key": "center", "label": "Abstract center", "value": "U(1)", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "derivation": { "method": "The center is (Z3 × Z2 × U1)/Γd. The finite-component projection of Γd has order d, leaving 6/d connected components; the identity component is a circle.", "inputs": [ "source:conrad-pi1-lie", "Γd projects injectively to both the diagonal finite center and the U(1) factor." ] }, "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#representations-1", "subject": "group:standard-model-z6", "key": "representations-1", "label": "Matter and the embedded kernel", "value": "A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/6) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#representations-2", "subject": "group:standard-model-z6", "key": "representations-2", "label": "Matter and the embedded kernel", "value": "All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#topology-1", "subject": "group:standard-model-z6", "key": "topology-1", "label": "Topology and the universal cover", "value": "The universal cover is SU(3) × SU(2) × R. The lift of Γ_6 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_6. The compact product used as the quotient domain already has fundamental group Z.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#conventions-1", "subject": "group:standard-model-z6", "key": "conventions-1", "label": "Hypercharge and SU(5)", "value": "Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#relation-1", "subject": "group:standard-model-z6", "key": "relation-1", "label": "The common gauge algebra", "value": "algebra:standard-model-r", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-algebra", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#relation-2", "subject": "group:standard-model-z6", "key": "relation-2", "label": "Specified compact covering product", "value": "group:standard-model-z1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "central-quotient-of", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "group:standard-model-z6#relation-3", "subject": "group:standard-model-z6", "key": "relation-3", "label": "Block subgroup of SU(5) with the specified map", "value": "group:su-5", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "embeds-in", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z2#definition", "subject": "map:standard-model-cover-to-z2", "key": "definition", "label": "Definition", "value": "The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_2=⟨γ^3⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 2.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z2#kernel", "subject": "map:standard-model-cover-to-z2", "key": "kernel", "label": "Kernel", "value": "Γ_2 = ⟨γ^3⟩ ≅ Z_2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z2#injective", "subject": "map:standard-model-cover-to-z2", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z2#surjective", "subject": "map:standard-model-cover-to-z2", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z2#matter-1", "subject": "map:standard-model-cover-to-z2", "key": "matter-1", "label": "Matter descent", "value": "A representation of the compact product descends exactly when every generator of this embedded kernel acts trivially. Test the color center, weak center, and abelian phase together; the same abstract cyclic group embedded in one factor can give a different answer.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z2#relation-1", "subject": "map:standard-model-cover-to-z2", "key": "relation-1", "label": "Compact covering product", "value": "group:standard-model-z1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z2#relation-2", "subject": "map:standard-model-cover-to-z2", "key": "relation-2", "label": "Chosen global form", "value": "group:standard-model-z2", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z3#definition", "subject": "map:standard-model-cover-to-z3", "key": "definition", "label": "Definition", "value": "The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_3=⟨γ^2⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 3.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z3#kernel", "subject": "map:standard-model-cover-to-z3", "key": "kernel", "label": "Kernel", "value": "Γ_3 = ⟨γ^2⟩ ≅ Z_3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z3#injective", "subject": "map:standard-model-cover-to-z3", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z3#surjective", "subject": "map:standard-model-cover-to-z3", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z3#matter-1", "subject": "map:standard-model-cover-to-z3", "key": "matter-1", "label": "Matter descent", "value": "A representation of the compact product descends exactly when every generator of this embedded kernel acts trivially. Test the color center, weak center, and abelian phase together; the same abstract cyclic group embedded in one factor can give a different answer.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z3#relation-1", "subject": "map:standard-model-cover-to-z3", "key": "relation-1", "label": "Compact covering product", "value": "group:standard-model-z1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z3#relation-2", "subject": "map:standard-model-cover-to-z3", "key": "relation-2", "label": "Chosen global form", "value": "group:standard-model-z3", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z6#definition", "subject": "map:standard-model-cover-to-z6", "key": "definition", "label": "Definition", "value": "The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_6=⟨γ^1⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 6.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z6#kernel", "subject": "map:standard-model-cover-to-z6", "key": "kernel", "label": "Kernel", "value": "Γ_6 = ⟨γ^1⟩ ≅ Z_6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z6#injective", "subject": "map:standard-model-cover-to-z6", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z6#surjective", "subject": "map:standard-model-cover-to-z6", "key": "surjective", "label": "Surjective", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z6#matter-1", "subject": "map:standard-model-cover-to-z6", "key": "matter-1", "label": "Matter descent", "value": "A representation of the compact product descends exactly when every generator of this embedded kernel acts trivially. Test the color center, weak center, and abelian phase together; the same abstract cyclic group embedded in one factor can give a different answer.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z6#relation-1", "subject": "map:standard-model-cover-to-z6", "key": "relation-1", "label": "Compact covering product", "value": "group:standard-model-z1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-cover-to-z6#relation-2", "subject": "map:standard-model-cover-to-z6", "key": "relation-2", "label": "Chosen global form", "value": "group:standard-model-z6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#definition", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "definition", "label": "Definition", "value": "The homomorphism ι(A,B,z)=diag(z² A,z⁻³ B) from SU(3) × SU(2) × U(1) to SU(5), with the color block first. Its kernel is generated by (exp(−2πi/3) I₃,−I₂,exp(2πi/6)) and its image is S(U(3) × U(2)).", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#kernel", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "kernel", "label": "Kernel", "value": "Z6 with the explicitly stated generator", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#image", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "image", "label": "Image", "value": "S(U(3) × U(2))", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#injective", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "injective", "label": "Injective", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#surjective", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "surjective", "label": "Surjective onto SU(5)", "value": "No", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#normalization", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "normalization", "label": "Physical hypercharge", "value": "Y=−X/6; integer Q=6Y=−X", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#branching-1", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "branching-1", "label": "Keep the charge", "value": "The same nonabelian multiplet can occur at different U(1) charges. The complete output is labeled by both SU highest weights and X. The kernel condition is −2t3+3t2+X=0 mod 6. The tool separately displays Y=−X/6 and transfers Q=−X when opening the product-group calculation.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#global-form-1", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "global-form-1", "label": "From homomorphism to embedding", "value": "The map is not injective on the compact product. Quotienting by its actual kernel gives the block subgroup. Inverting the U(1) coordinate changes from the X convention to Q=6Y and identifies this quotient with the G_SM^(6) record.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#relation-1", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "relation-1", "label": "Underlying compact product, with the X coordinate stated here", "value": "group:standard-model-z1", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#relation-2", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "relation-2", "label": "Parent group", "value": "group:su-5", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:su-3-su-2-u1-to-su-5-block#relation-3", "subject": "map:su-3-su-2-u1-to-su-5-block", "key": "relation-3", "label": "Image after converting X to Q=−X", "value": "group:standard-model-z6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "image-isomorphic-to", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-z6-into-su5#definition", "subject": "map:standard-model-z6-into-su5", "key": "definition", "label": "Definition", "value": "The map [(A,B,z)] ↦ diag(z⁻² A,z³ B) from G_SM^(6) to SU(5). The numerator map has precisely the diagonal Γ6 kernel, so it descends to an injective homomorphism with image S(U(3) × U(2)).", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-z6-into-su5#kernel", "subject": "map:standard-model-z6-into-su5", "key": "kernel", "label": "Kernel", "value": "Trivial", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-z6-into-su5#injective", "subject": "map:standard-model-z6-into-su5", "key": "injective", "label": "Injective", "value": "Yes", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-z6-into-su5#image", "subject": "map:standard-model-z6-into-su5", "key": "image", "label": "Image", "value": "S(U(3) × U(2))", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-z6-into-su5#physics-1", "subject": "map:standard-model-z6-into-su5", "key": "physics-1", "label": "What the embedding establishes", "value": "The map provides the global group and charge conventions for SU(5) branching. It does not establish that the observed Standard Model is unified in SU(5), or select a Higgs potential, symmetry-breaking vacuum, or ultraviolet dynamics.", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-z6-into-su5#relation-1", "subject": "map:standard-model-z6-into-su5", "key": "relation-1", "label": "The specified diagonal quotient", "value": "group:standard-model-z6", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-domain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-z6-into-su5#relation-2", "subject": "map:standard-model-z6-into-su5", "key": "relation-2", "label": "Parent compact group", "value": "group:su-5", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "has-codomain", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" }, { "id": "map:standard-model-z6-into-su5#relation-3", "subject": "map:standard-model-z6-into-su5", "key": "relation-3", "label": "Computable block map in the X convention", "value": "map:su-3-su-2-u1-to-su-5-block", "evidence": [ { "source": "block-baez-huerta", "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly." }, { "source": "tong", "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices" }, { "source": "conrad-pi1-lie", "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup" } ], "assumptions": [], "relationship": "related-map", "reviewStatus": "pending-expert-review", "revision": "2026.11-workbench.4" } ], "capabilities": [ { "id": "global-forms", "title": "Global forms & representations", "description": "Change the global group. See which highest weights descend.", "category": "Finite-dimensional ordinary complex representations where applicable", "scope": "SU(2), SO(3), SU(3), PSU(3)", "symbol": "G / Γ", "version": "groups-design-1.0.0", "bounds": "SU(2), SO(3), SU(3), PSU(3); each highest-weight input 0–30. Complete dimension-bounded browser: A1 up to 30, A2 up to 100." }, { "id": "tensor-products", "title": "Tensor products", "description": "Decompose two SU(2) representations with exact multiplicities.", "category": "Finite-dimensional ordinary complex representations where applicable", "scope": "SU(2), highest weights 0–30", "symbol": "R ⊗ S", "version": "groups-design-1.0.0", "bounds": "Two SU(2) highest-weight inputs 0–30; output highest weights can reach 60." }, { "id": "invariants", "title": "Invariant couplings", "description": "Compare distinct fields with one identical commuting scalar.", "category": "Finite-dimensional ordinary complex representations where applicable", "scope": "A₄ triplet, degree 0–12", "symbol": "Inv(R)", "version": "groups-design-1.0.0", "bounds": "The A4 real triplet, tensor or symmetric powers, degrees 0–12. No fermion or derivative operator engine." }, { "id": "branching", "title": "Branching & embeddings", "description": "Follow the regular or principal map into SU(3).", "category": "Finite-dimensional ordinary complex representations where applicable", "scope": "SU(3) fundamental and adjoint", "symbol": "G → H", "version": "groups-design-1.0.0", "bounds": "Only SU(3) representations 3 and 8 along the named regular and principal maps." }, { "id": "topology", "title": "Symmetry breaking & topology", "description": "Explore the homotopy sequence for two spherical vacuum orbits.", "category": "Finite-dimensional ordinary complex representations where applicable", "scope": "SU(2)/U(1) and SO(3)/SO(2)", "symbol": "G / H", "version": "groups-design-1.0.0", "bounds": "Only the specified SU(2)/U(1) and SO(3)/SO(2) fibrations with declared angular conventions." }, { "id": "finite-characters", "title": "Finite characters & tensor products", "symbol": "χ", "category": "Finite groups", "description": "Decompose tensor, symmetric, and exterior powers using exact ordinary complex characters.", "scope": "C₃, S₃, D₈, Q₈, A₄, S₄, A₅; characteristic zero.", "bounds": "Seven curated groups; base representations of complex dimension ≤60; multiplicities ≤12; powers of degree 0–12.", "version": "finite-characters-1.0.0" }, { "id": "lie-representations", "title": "Highest weights & Weyl dimensions", "symbol": "λ", "category": "Compact Lie groups", "description": "Calculate exact dimensions and quadratic Casimirs from a highest weight, with the root conventions visible.", "scope": "Simply connected compact forms of the selected simple Lie types.", "bounds": "A₁–A₈, B₂–B₈, C₂–C₈, D₄–D₈, E₆, E₇, E₈, F₄, G₂; Dynkin labels 0–30. No noncompact unitary-dual calculation.", "version": "lie-representations-1.0.0" }, { "id": "su3-tensor-products", "title": "SU(3) tensor products", "symbol": "⊗", "category": "Compact Lie groups", "description": "Resolve a product into irreducible highest weights, retaining conjugates and repeated multiplets.", "scope": "Finite-dimensional continuous complex SU(3) representations.", "bounds": "Each Dynkin label of either input is 0–6. Complete decomposition within that input domain; output labels may be larger.", "version": "lie-representations-1.0.0" }, { "id": "su3-branching", "title": "SU(3) branching by a chosen map", "symbol": "↘", "category": "Symmetry breaking", "description": "Restrict any supported SU(3) highest weight along the regular or principal SU(2) map.", "scope": "Regular SU(2) inclusion or principal map with SO(3) image.", "bounds": "SU(3) labels p,q from 0–6. Complete SU(2) decomposition; the principal map has kernel Z₂.", "version": "lie-representations-1.0.0" }, { "id": "matter-quotients", "title": "Matter & global gauge structure", "symbol": "Z", "category": "Gauge theory", "description": "Find the common central kernel of matter multiplets and every compatible central quotient of SU(N).", "scope": "One compact SU(N) factor; finite-dimensional continuous complex matter representations.", "bounds": "2≤N≤12; at most 16 multiplets; each Dynkin label 0–30. Kinematic descent only.", "version": "su-n-matter-kernel-1.0.0" }, { "id": "sun-tensor-products", "title": "SU(N) tensor products", "symbol": "⊗", "category": "Model building", "description": "Decompose compact SU(N) representations with exact Littlewood–Richardson coefficients and explicit Young-diagram conventions.", "scope": "Finite-dimensional continuous complex SU(N) representations; 2≤N≤6.", "bounds": "Each Dynkin label 0–12; at most 16 canonical Young boxes across the two inputs. Complete decomposition within the supported domain.", "version": "su-n-tensors-1.0.0" }, { "id": "charged-branching", "title": "Branching with U(1) charges", "symbol": "↘", "category": "Symmetry breaking", "description": "Restrict an SU(N) representation along a specified block map, keeping both nonabelian labels and the commuting U(1) charge.", "scope": "SU(p) × SU(N−p) × U(1) mapped into SU(N); 3≤N≤6, 1≤p