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All 93 GAP fixtures were generated and serialization-checked; native GAP execution remains unverified. Results concern those recorded fixtures, not arbitrary future downloads." }, "methods": [ { "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. 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No noncompact unitary-dual calculation.", "version": "lie-representations-1.0.0", "algorithmVersion": "lie-representations-1.0.0", "route": "/tools/lie-representations/", "conventionProfileIds": [ "lie-bourbaki-long-root-2-v1" ], "method": "Weyl dimension product over positive coroots; inverse-Cartan weight pairing for C2 and trace index; exact root-lattice test for descent to the adjoint group; conjugation and self-dual reality indicator.", "sourceIds": [ "workbench-etingof-lie", "workbench-sage-roots", "workbench-sage-e", "workbench-sage-b", "workbench-sage-c", "workbench-sage-f", "workbench-sage-g", "sage", "workbench-sage-gt", "haber", "conrad-centers", "workbench-adams-indicator" ], "dataResources": [ "/data/root-systems.json" ], "reproduction": { "language": "SageMath", "extension": ".py", "scope": "Reconstruct the stated root system and verify saved Weyl invariants." }, "checks": [ "Positive-coroot counts", "Classical defining/exterior/spin dimensions", "Exceptional fundamental and adjoint fixtures", "Exact dimension integrality", "Trace identity", "Full-center descent and reality distinctions" ] }, { "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", "algorithmVersion": "lie-representations-1.0.0", "route": "/tools/su3-tensor-products/", "conventionProfileIds": [ "su3-dynkin-pq-v1", "lie-bourbaki-long-root-2-v1" ], "method": "Gelfand–Tsetlin weight characters, exact multiplication, and highest-weight subtraction until the complete product character is reconstructed.", "sourceIds": [ "workbench-etingof-lie", "sage", "workbench-sage-gt" ], "reproduction": { "language": "SageMath", "extension": ".py", "scope": "Reconstruct the SU(3) product and compare complete Weyl characters." }, "checks": [ "Complete character reconstruction", "Dimension conservation", "Nonnegative integral multiplicities", "Conjugation and triality", "Pieri and low-weight fixtures" ] }, { "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", "algorithmVersion": "lie-representations-1.0.0", "route": "/tools/su3-branching/", "conventionProfileIds": [ "su3-dynkin-pq-v1", "su3-regular-su2-v1", "su3-principal-su2-v1" ], "method": "Restrict the complete Gelfand–Tsetlin character along the specified torus map; recover SU(2) highest-weight multiplicities by c_n − c_(n+2).", "sourceIds": [ "workbench-etingof-lie", "sage", "workbench-sage-gt", "haber" ], "maps": [ { "id": "regular", "mapId": "map:su-2-to-su-3-regular", "name": "Regular block inclusion", "domain": "SU(2)", "codomain": "SU(3)", "image": "SU(2)", "kernel": "1", "injective": true, "formula": "U ↦ diag(U,1)", "torusFormula": "diag(z,z^-1) ↦ diag(z,z^-1,1)", "torusLatex": "\\operatorname{diag}(z,z^{-1})\\longmapsto\\operatorname{diag}(z,z^{-1},1)", "embeddingIndex": 1, "conventionId": "su3-regular-su2-v1" }, { "id": "principal", "mapId": "map:su-2-to-su-3-principal", "name": "Principal spin-one homomorphism", "domain": "SU(2)", "codomain": "SU(3)", "image": "SO(3)", "kernel": "{±I}", "injective": false, "formula": "U ↦ Sym²(U)", "torusFormula": "diag(z,z^-1) ↦ diag(z^2,1,z^-2)", "torusLatex": "\\operatorname{diag}(z,z^{-1})\\longmapsto\\operatorname{diag}(z^2,1,z^{-2})", "embeddingIndex": 4, "conventionId": "su3-principal-su2-v1" } ], "checks": [ "Complete restricted-character reconstruction", "Dimension conservation", "Even SU(2) labels under the principal map", "Independent symmetric-fundamental fixtures" ] }, { "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", "algorithmVersion": "su-n-matter-kernel-1.0.0", "route": "/tools/matter-quotients/", "conventionProfileIds": [ "su-n-dynkin-center-v1" ], "method": "q = sum(i*a_i) mod N; common central kernel has order gcd(N,q_1,...,q_s); all divisors of that order define compatible central quotients. 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Complete decomposition within the supported domain.", "version": "su-n-tensors-1.0.0", "algorithmVersion": "su-n-tensors-1.0.0", "route": "/tools/sun-tensor-products/", "conventionProfileIds": [ "su-n-young-lr-v1" ], "method": "Complete Littlewood–Richardson skew-tableau enumeration for canonical polynomial GL(N) lifts; remove determinant columns only on restriction to SU(N). Retain original GL partitions for an independent exact alternant specialization.", "sourceIds": [ "workbench-lrcalc", "workbench-stanley-lr", "workbench-sage-schur", "sage", "conrad-centers", "workbench-adams-indicator", "workbench-lieart" ], "limits": { "minN": 2, "maxN": 6, "maxDynkinLabel": 12, "maxCombinedBoxes": 16, "maxTableauNodes": 2000000 }, "checks": [ "Exact dimension conservation", "Nonnegative integral multiplicities", "Center-charge conservation", "Exact Schur-character specialization, with determinant factors retained", "Dual singlet condition", "Independent SU(3), Pieri, exterior-power, and published fixtures" ], "representationCategory": "Finite-dimensional continuous complex representations of compact simply connected SU(N), characteristic zero.", "physicalScope": "Abstract irreducible summands and multiplicities; no normalized Clebsch–Gordan basis, general exchange projections, or full physical operator enumeration.", "dataResources": [ "/data/model-setups/su5-ten-tensor.json" ] }, { "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 diag(z^(q/g)A,z^(-p/g)B), with g=gcd(p,q). Retain both block determinant powers until assigning X=(q|mu|-p|nu|)/g; only then use SU Dynkin labels.", "sourceIds": [ "workbench-lrcalc", "branching", "block-baez-huerta", "tong" ], "limits": { "minN": 3, "maxN": 6, "minBlockSize": 1, "maxDynkinLabel": 12, "maxCanonicalBoxes": 12 }, "map": { "domain": "SU(p) × SU(q) × U(1)", "codomain": "SU(p+q)", "image": "S(U(p) × U(q))", "u1Period": "2π", "kernel": "Cyclic of order pq/g; generator (exp(-2πi/p) I_p, exp(2πi/q) I_q, exp(2πi/(pq/g))). SU(1) is the trivial factor." }, "hyperchargeConvention": "For N=5 and p=3, physical Y=-X/6 with Qelectric=T3+Y. Product-gauge input uses Q=6Y=-X; the cross-tool transfer changes the integer charge sign.", "checks": [ "Complete dimension conservation", "Descent through the specified map kernel", "Vanishing first charge moment", "Exact quadratic charge trace from the parent index", "Independent Jacobi–Trudi character evaluations", "Determinant-shift independence, conjugation, and block exchange" ], "physicalScope": "A specified regular block restriction; no arbitrary embedding search, normalized intertwiner, vacuum selection, or mass spectrum.", "dataResources": [ "/data/model-setups/su5-ten-branching.json" ] }, { "id": "product-gauge-groups", "title": "Product gauge groups & matter", "symbol": "Γ", "category": "Global gauge structure", "description": "Find the common central kernel of matter, then test an explicitly generated quotient of a product gauge group.", "scope": "One or two SU(N) factors, 2≤N≤8, and one compact U(1) with period 2π.", "bounds": "At most 16 multiplets; Dynkin labels 0–12; integer U(1) charges −24…24; at most 3 quotient generators with phase denominators ≤60; generated subgroup order ≤4096.", "version": "product-gauge-1.0.0", "algorithmVersion": "product-gauge-1.0.0", "route": "/tools/product-gauge-groups/", "conventionProfileIds": [ "product-su-u1-center-v1" ], "method": "Exact center-phase congruences for matter in SU(N1) [× SU(N2)] × U(1); finite subgroup closure from explicitly embedded generators; common-kernel enumeration, invariant factors, and the universal-cover lift of the chosen quotient.", "sourceIds": [ "conrad-centers", "tong", "conrad-pi1-lie", "ast" ], "limits": { "minSUFactors": 1, "maxSUFactors": 2, "minFactorSize": 2, "maxFactorSize": 8, "maxMatterMultiplets": 16, "maxDynkinLabel": 12, "minU1Charge": -24, "maxU1Charge": 24, "maxGenerators": 3, "maxGeneratorPhaseDenominator": 60, "maxGeneratedSubgroupOrder": 4096 }, "inputConvention": "The input key ranks contains SU factor sizes N_i, not Lie-algebra ranks N_i-1. Integer U(1) charges use a circle of period 2π. A quotient generator supplies all nonabelian center exponents and a rational U(1) phase in turns.", "standardModelConvention": "Color precedes weak isospin. Q=6Y and gamma=(omega3,-I2,exp(2πi/6)). Compatible quotient orders are 1,2,3,6; the actual SU(5) block subgroup uses the full diagonal Z6.", "checks": [ "Exact matter phases", "Closure and order of the embedded subgroup", "Per-multiplet descent obstructions", "Finite and continuous common kernels", "Invariant factors including noncyclic cases", "Universal-cover lift and fundamental-group torsion" ], "physicalScope": "Kinematic representation descent and topology of the supplied compact quotient; no anomaly cancellation, full line-operator spectrum, theta choice, or dynamical vacuum.", "dataResources": [ "/data/model-setups/standard-model-z6.json" ] }, { "id": "local-operators", "title": "Local operators & field statistics", "symbol": "𝒪", "category": "Fields and interactions", "description": "Count independent Lorentz and gauge singlets for a specified field monomial, with Bose/Fermi statistics acting on the complete field representation.", "scope": "Four-dimensional Lorentzian scalar and Weyl fields; SU(3) × SU(2) × U(1) or one of its specified diagonal quotients; complex polynomials without derivatives.", "bounds": "At most 16 species, 3 flavors per species, 6 field insertions, and canonical dimension 6. Color labels p+q≤2, weak label 0–2, integer charges −24…24. Exact completion or an explicit resource-limit error.", "version": "local-operators-1.0.0", "algorithmVersion": "local-operators-1.0.0", "route": "/tools/local-operators/", "conventionProfileIds": [ "four-d-local-polynomial-v1", "product-su-u1-center-v1" ], "method": "Build exact SU(3), weak SU(2), and complexified Lorentz-spin characters for each complete field representation, including ungauged flavor multiplicity. Newton identities compute symmetric/exterior characters at the requested power. The positive-root Weyl denominator extracts the trivial representation after tensoring species. Enforce integral U(1) charge neutrality and global descent separately.", "sourceIds": [ "marinissen-rahn-waalewijn-2020", "lehman-martin-2015", "fonseca-2020", "dreiner-haber-martin-2010", "henning-lu-melia-murayama-2017", "weinberg-1979-lepton-number", "conrad-centers", "tong", "conrad-pi1-lie", "ast" ], "limits": { "maxFields": 16, "maxFieldIdLength": 24, "maxFieldNameLength": 48, "maxSU3LabelSum": 2, "maxSU2Label": 2, "minCharge": -24, "maxCharge": 24, "maxFlavors": 3, "maxPower": 6, "maxTotalDegree": 6, "maxCanonicalDimension": 6, "maxStates": 120000, "maxOperations": 20000000, "resourcePolicy": "At most 120000 distinct weights in any intermediate character and 20000000 coefficient multiplication pairs per request. Exceeding either bound throws without returning a partial count." }, "representationCategory": "Finite-dimensional complex scalar and Weyl field representations in four-dimensional Lorentzian spacetime, ordinary compact gauge representations, and polynomial/exterior component algebras at one fixed multidegree.", "physicalScope": "Exact complex invariant dimension, with all algebraic index identities included. Flavor is ungauged multiplicity. No derivatives, field strengths, EOM, IBP, Hermiticity identification, flavor-basis quotient, anomaly check, automatic full EFT enumeration, or general explicit contraction basis.", "checks": [ "Component-space dimension identities", "Nonnegative integral counts", "Graded count bounded by distinguishable-slot count", "Weyl invariance where projection is performed", "Independent explicit component enumeration and full-root Haar projection", "Independent exact raising-operator kernels", "Weinberg and QQQL flavor polynomials", "All declared fields descend to the selected global quotient" ], "dataResources": [ "/data/physical-setups.json", "/data/physical-setup.schema.json" ], "setupSchema": "groups.qft.org/physical-setup/1", "calculationSchema": "groups.qft.org/physical-calculation/1" }, { "id": "operator-enumeration", "title": "Enumerate allowed interactions", "symbol": "ℋ", "category": "Fields and interactions", "description": "Enumerate all permitted scalar and Weyl monomials through a chosen dimension, then count their exact Lorentz and gauge invariants with field statistics.", "scope": "Four-dimensional Lorentzian scalar and Weyl inventories; SU(3) × SU(2) × U(1) and the specified diagonal quotients; optional integer U(1) and cyclic selection rules; no derivatives.", "bounds": "At most 16 species, 3 flavors per species, canonical dimension 6, and 3 additional charge rules. Color labels p+q≤2, weak label 0–2, integer gauge and selection charges −24…24. Explicit resource bounds; complete results or an error.", "version": "operator-enumeration-1.0.0", "algorithmVersion": "operator-enumeration-1.0.0", "route": "/tools/operator-enumeration/", "conventionProfileIds": [ "four-d-local-polynomial-v1", "product-su-u1-center-v1" ], "method": "Enumerate all bounded occupation vectors at the selected canonical dimensions; test global descent for every field; reject impossible gauge charges, optional U(1)/cyclic selections, Lorentz parity, center charges, and exterior capacities; project every surviving complete graded character exactly. Sum independent multidegree dimensions without identifying conjugates.", "sourceIds": [ "marinissen-rahn-waalewijn-2020", "lehman-martin-2015", "fonseca-2020", "dreiner-haber-martin-2010", "henning-lu-melia-murayama-2017", "alonso-jenkins-manohar-trott-2014-counting", "alonso-et-al-2014-baryon-violation", "kobach-2016-dimension-selection" ], "limits": { "maxFields": 16, "maxDimension": 6, "maxSymmetries": 3, "maxModulus": 12, "maxAbsoluteCharge": 24, "maxVisited": 100000, "maxCandidates": 4096, "maxOperations": 100000000, "maxOutputRows": 4096, "maxBytes": 4194304 }, "representationCategory": "Complex scalar and Weyl polynomial/exterior algebras in four-dimensional Lorentzian spacetime, with explicit compact gauge representations and ungauged flavor multiplicity.", "physicalScope": "Complete enumeration within the stated inventory, dimension range, and selection rules. No derivatives, field strengths, EOM, IBP, Hermiticity or flavor-basis identifications, anomaly checks, or generated normalized contraction bases.", "checks": [ "Independent Cartesian occupation enumeration without selection pruning", "Scalar binomial Hilbert-series coefficients", "Dimension-range disjoint unions", "Global charge-sector partitions", "Conjugation symmetry", "One- and three-generation Standard Model sector benchmarks", "Explicit all-field descent", "Strict file replay and resource refusals" ], "dataResources": [ "/data/operator-inventories.json", "/data/operator-enumeration.schema.json" ], "setupSchema": "groups.qft.org/operator-enumeration/1", "calculationSchema": "groups.qft.org/operator-enumeration-calculation/1" } ], "conventions": "/data/conventions.json", "sources": [ { "id": "haber", "title": "Haber · Spin covers and subgroup examples", "authors": [ "Howard E. 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Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "sage", "title": "SageMath · Weyl characters", "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 source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "branching", "title": "SageMath · Branching rules", "authors": [ "Daniel Bump", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/branching_rules.html", "kind": "documentation", "locatorHint": "Branching rules and embeddings of maximal tori: restriction depends on the specified map.", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "gap", "title": "GAP · Characters and symmetrizations", "authors": [], "url": "https://gap-system.github.io/gap/doc/ref/chap72_mj.html", "kind": "documentation", "locatorHint": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "a4", "title": "GroupNames · A₄", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/A4.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "ast", "title": "Aharony, Seiberg and Tachikawa · Global gauge structure", "authors": [ "Ofer Aharony", "Nathan Seiberg", "Yuji Tachikawa" ], "url": "https://arxiv.org/abs/1305.0318", "kind": "paper", "locatorHint": "§1.1 and §2: global group, allowed representations, and additional line-operator choices", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "tong", "title": "Tong · Line operators in the Standard Model", "authors": [ "David Tong" ], "url": "https://arxiv.org/abs/1705.01853", "kind": "paper", "locatorHint": "Introduction and Section 3: the Z6 quotient, integer-normalized hypercharge, and the additional choice of line operators.", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "groupnames-c-3", "title": "GroupNames: C₃", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C3.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "groupnames-s-3", "title": "GroupNames: S₃", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/S3.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "groupnames-s-4", "title": "GroupNames: S₄", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/S4.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "groupnames-a-5", "title": "GroupNames: A₅", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/A5.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "groupnames-q-8", "title": "GroupNames: Q₈", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/Q8.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "id": "groupnames-d-8", "title": "GroupNames: D₈", "authors": [ "Tim Dokchitser" ], "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/D4.html", "kind": "database", "locatorHint": "Order, Centre, Aliases, generators and relations, Character table", "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted." }, { "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": "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. 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-b", "title": "SageMath: Root system data for type B", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_B.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(): final node is short.", "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed." }, { "id": "workbench-sage-c", "title": "SageMath: Root system data for type C", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/type_C.html", "kind": "documentation", "locatorHint": "AmbientSpace.simple_root(): final node is long.", "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-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-gt", "title": "SageMath: Gelfand–Tsetlin patterns", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/gelfand_tsetlin_patterns.html", "kind": "documentation", "locatorHint": "GelfandTsetlinPattern.weight(): interlacing arrays and differences of row sums.", "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." }, { "id": "cohen-center", "title": "Thomas D. Cohen, Center symmetry and area laws", "authors": [], "url": "https://arxiv.org/abs/1407.4128", "kind": "paper", "locatorHint": "Property (i), equation (3), and the argument using N-ality and the greatest common divisor", "licenseNote": "Linked primary source supporting original mathematical descriptions and exact calculations. Upstream text and software are not redistributed; consult the source for its own reuse terms." }, { "id": "workbench-lrcalc", "title": "SageMath: Littlewood–Richardson Calculator interface", "authors": [ "Anders Buch", "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/libs/sage/libs/lrcalc/lrcalc.html", "kind": "documentation", "locatorHint": "lrcoef, mult, coprod: coefficients, Schur products, and coproducts; the [2,1] square fixture.", "licenseNote": "Linked primary mathematical or software documentation supporting original algorithms and prose. No upstream implementation or document text is redistributed." }, { "id": "workbench-stanley-lr", "title": "Symmetries of Plane Partitions", "authors": [ "Richard P. Stanley" ], "url": "https://math.mit.edu/~rstan/pubs/pubfiles/65.pdf", "kind": "paper", "locatorHint": "Section 3, Theorem 3.2: skew-tableau formulation of the Littlewood–Richardson rule.", "licenseNote": "Linked primary mathematical or software documentation supporting original algorithms and prose. No upstream implementation or document text is redistributed." }, { "id": "workbench-sage-schur", "title": "SageMath: Schur symmetric functions", "authors": [ "SageMath developers" ], "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/schur.html", "kind": "documentation", "locatorHint": "Schur bases, product and coproduct, and specialization as polynomial characters.", "licenseNote": "Linked primary mathematical or software documentation supporting original algorithms and prose. 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. Martin" ], "url": "https://arxiv.org/abs/0812.1594", "kind": "paper", "locatorHint": "§2 and §3.2: two-component Lorentz spinors, spinor contractions and symmetric fermion mass matrices.", "licenseNote": "Linked primary reference; no upstream text or figures are mirrored." }, { "id": "henning-lu-melia-murayama-2017", "title": "Higher dimension operators in the SM EFT", "authors": [ "Brian Henning", "Xiaochuan Lu", "Tom Melia", "Hitoshi Murayama" ], "url": "https://arxiv.org/abs/1512.03433", "kind": "paper", "locatorHint": "§2: operator Hilbert series and the additional treatment required for derivatives, equations of motion and integration by parts.", "licenseNote": "Linked primary research paper; used to delimit the scope of this derivative-free calculation." }, { "id": "weinberg-1979-lepton-number", "title": "Steven Weinberg, Baryon- and Lepton-Nonconserving Processes", "authors": [ "Steven Weinberg" ], "url": "https://doi.org/10.1103/PhysRevLett.43.1566", "kind": "paper", "locatorHint": "Phys. Rev. Lett. 43, 1566–1570 (1979), Eq. (20): the dimension-five lepton-number-violating interaction", "licenseNote": "Linked primary research paper. The local worked contraction is an original expression in the site’s stated spinor and charge conventions." } ], "independentExpertReview": "pending", "modelCalculationFiles": { "schema": "groups.qft.org/model-calculation/1", "resources": "/data/model-setups.json", "scope": "Named single-tool inputs, exact results, conventions, sources, and versions for the three Research workbench II tools. Loading restores inputs and recomputes with the current engine. This is not the complete proposed PhysicalSetup schema." }, "operatorEnumerationFiles": { "setupSchema": "groups.qft.org/operator-enumeration/1", "calculationSchema": "groups.qft.org/operator-enumeration-calculation/1", "resources": "/data/operator-inventories.json", "jsonSchema": "/data/operator-enumeration.schema.json", "scope": "A bounded field inventory, dimension range, optional identity, and up to three continuous or cyclic charge selections. Imports recompute every permitted monomial; conjugates remain explicit." }, "physicalSetupFiles": { "setupSchema": "groups.qft.org/physical-setup/1", "calculationSchema": "groups.qft.org/physical-calculation/1", "resources": "/data/physical-setups.json", "jsonSchema": "/data/physical-setup.schema.json", "scope": "Four-dimensional Lorentzian scalar/Weyl fields and a fixed derivative-free monomial. Every declared field must descend; all imports validate and recompute the supplied inputs. A bounded physical setup is implemented; the broader proposed physical model remains future work." }, "coverage": "Research workbench IV adds exhaustive bounded field-inventory enumeration and charge selection to the fourteen inherited tools: fifteen tools in total. Derivatives, field strengths, EOM/IBP reduction, and explicit contraction bases remain future work." }