{
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      "title": "Zuber · Invariances in Physics and Group Theory",
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    },
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    },
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    },
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        "Brian Conrad",
        "Aaron Landesman"
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      "kind": "paper",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
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      "kind": "paper",
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      "authors": [],
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      "kind": "paper",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "eft",
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      "kind": "paper",
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    {
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    },
    {
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    },
    {
      "id": "groupnames-q-8",
      "title": "GroupNames: Q₈",
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      "kind": "database",
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      "id": "groupnames-d-8",
      "title": "GroupNames: D₈",
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      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "finite-wilson-introduction",
      "title": "Finite simple groups: Lecture 1 — Introduction and alternating groups",
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        "Robert A. Wilson"
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      "kind": "lecture-notes",
      "locatorHint": "CFSG statement and alternating groups, slides 1–4",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
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    {
      "id": "finite-wilson-linear",
      "title": "Finite simple groups: Lecture 2 — Linear groups",
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      "kind": "lecture-notes",
      "locatorHint": "Definition, orders and simplicity of PSL",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
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      "id": "finite-wilson-classical",
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      "kind": "lecture-notes",
      "locatorHint": "Symplectic, unitary and orthogonal groups",
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      "id": "finite-wilson-classical-notes",
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        "Robert A. Wilson"
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      "url": "https://webspace.maths.qmul.ac.uk/r.a.wilson/FSG/notes3.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Sections 3.2–3.5",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-wilson-exceptional",
      "title": "Finite simple groups: Lecture 4 — Exceptional groups",
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      "kind": "lecture-notes",
      "locatorHint": "Ten exceptional families; Suzuki–Ree parameters and small cases; G2",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-wilson-suzuki-ree",
      "title": "On the simple groups of Suzuki and Ree",
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        "Robert A. Wilson"
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      "kind": "paper",
      "locatorHint": "Construction and simplicity, especially section 5",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-gap-groups",
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        "The GAP Group"
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      "url": "https://gap-system.github.io/gap/doc/ref/chap39_mj.html",
      "kind": "documentation",
      "locatorHint": "IsSimpleGroup; IsomorphismTypeInfoFiniteSimpleGroup; SimpleGroup",
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    {
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      "title": "GAP Reference Manual: Group Libraries",
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        "The GAP Group"
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      "url": "https://gap-system.github.io/gap/doc/ref/chap50_mj.html",
      "kind": "documentation",
      "locatorHint": "Basic and classical group constructors",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-gap-classical-maximals",
      "title": "GAP ClassicalMaximals: Classical Groups and Aschbacher’s Theorem",
      "authors": [
        "GAP ClassicalMaximals contributors"
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      "url": "https://gap-packages.github.io/ClassicalMaximals/doc/chap2_mj.html",
      "kind": "documentation",
      "locatorHint": "Section 2.2-4: odd-dimensional orthogonal groups, including even characteristic",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
    },
    {
      "id": "finite-taylor-mckay",
      "title": "Character Theory of Finite Groups: Day 4 — The McKay Correspondence",
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      "kind": "lecture-notes",
      "locatorHint": "Binary polyhedral groups and the SU(2) → SO(3) cover, slides 7–8",
      "licenseNote": "Linked for verification. No redistribution licence is asserted; this dataset contains original summaries and selected mathematical facts, not source tables."
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    {
      "id": "finite-craven-identifications",
      "title": "Groups with a p-element acting with a single non-trivial Jordan block on a simple module in characteristic p",
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      "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",
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        "Bo Feng",
        "Amihay Hanany",
        "Yang-Hui He"
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      "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",
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        "Robert A. Wilson",
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      "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."
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    {
      "id": "finite-groupnames-c-2",
      "title": "GroupNames: C₂",
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      "kind": "database",
      "locatorHint": "Order, Centre, Aliases, generators and relations, Character table",
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      "kind": "database",
      "locatorHint": "Order, Centre, Aliases, generators and relations, Character table",
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      "title": "GroupNames: SL(2,5), binary icosahedral",
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      "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."
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    {
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      "title": "GroupNames: GL(3,2) ≅ PSL(2,7)",
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      "kind": "database",
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    {
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      "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."
    },
    {
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      "kind": "database",
      "locatorHint": "Order, Centre, Aliases, generators and relations, Character table",
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    },
    {
      "id": "finite-atlas-m11",
      "title": "ATLAS: Mathieu group M11",
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      "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."
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    {
      "id": "finite-atlas-m12",
      "title": "ATLAS: Mathieu group M12",
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      "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",
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      "id": "finite-atlas-m23",
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      "id": "finite-atlas-m24",
      "title": "ATLAS: Mathieu group M24",
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    {
      "id": "finite-atlas-j1",
      "title": "ATLAS: First Janko group",
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    },
    {
      "id": "finite-atlas-j2",
      "title": "ATLAS: Hall–Janko group",
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    {
      "id": "finite-atlas-j3",
      "title": "ATLAS: Third Janko group",
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      "locatorHint": "Heading; Order; Standard generators; Representations",
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    {
      "id": "finite-atlas-j4",
      "title": "ATLAS: Fourth Janko group",
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      "id": "finite-atlas-co1",
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      "id": "finite-atlas-co2",
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      "id": "finite-atlas-fi22",
      "title": "ATLAS: Fischer group Fi22",
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    {
      "id": "finite-atlas-fi23",
      "title": "ATLAS: Fischer group Fi23",
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    {
      "id": "finite-atlas-fi24-prime",
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      "kind": "database",
      "locatorHint": "Heading; Order; Standard generators; Representations",
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    {
      "id": "finite-atlas-hs",
      "title": "ATLAS: Higman–Sims group",
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      "kind": "database",
      "locatorHint": "Heading; Order; Standard generators; Representations",
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    },
    {
      "id": "finite-atlas-mcl",
      "title": "ATLAS: McLaughlin group",
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      "kind": "database",
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    {
      "id": "finite-atlas-he",
      "title": "ATLAS: Held group",
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    {
      "id": "finite-atlas-ru",
      "title": "ATLAS: Rudvalis group",
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    },
    {
      "id": "finite-atlas-suz",
      "title": "ATLAS: Sporadic Suzuki group",
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    },
    {
      "id": "finite-atlas-on",
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    {
      "id": "finite-atlas-hn",
      "title": "ATLAS: Harada–Norton group",
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      "kind": "database",
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      "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",
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    {
      "id": "finite-atlas-th",
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      "id": "finite-atlas-b",
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    {
      "id": "finite-atlas-tits",
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    {
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      "title": "Lecture 17: Classification of Connected Compact and Complex Reductive Groups",
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    {
      "id": "garrett-isogenies",
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    {
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      "licenseNote": "Linked author-hosted paper. The group definitions and dimensions used here are factual summaries, not copied text."
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    {
      "id": "brodzki-sl2-cover",
      "title": "K-theory and limit-of-discrete-series for the universal cover of SL₂(R)",
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    {
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    {
      "id": "finite-gap-power-maps",
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      "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."
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    {
      "id": "workbench-etingof-lie",
      "title": "Lie Groups and Lie Algebras I",
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    {
      "id": "workbench-sage-roots",
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      "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."
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    {
      "id": "workbench-sage-e",
      "title": "SageMath: Root system data for type E",
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      "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
    },
    {
      "id": "workbench-sage-b",
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      "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",
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      "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
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    {
      "id": "workbench-sage-f",
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      "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
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    {
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      "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
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    {
      "id": "workbench-sage-gt",
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      "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."
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    {
      "id": "workbench-adams-indicator",
      "title": "The Real Chevalley Involution",
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      "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",
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      "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"
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      "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": [
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      "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."
    },
    {
      "id": "alonso-jenkins-manohar-trott-2014-counting",
      "title": "Renormalization Group Evolution of the Standard Model Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology",
      "authors": [
        "Rodrigo Alonso",
        "Elizabeth E. Jenkins",
        "Aneesh V. Manohar",
        "Michael Trott"
      ],
      "url": "https://arxiv.org/abs/1312.2014",
      "kind": "paper",
      "locatorHint": "Appendix A and Table 2: flavor-dependent coefficient counts, including 30 and 2205 baryon-preserving four-fermion directions for one and three generations.",
      "licenseNote": "Linked primary research paper. 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. Used as a necessary selection-rule cross-check, not as a replacement for exact invariant counting."
    }
  ]
}