{ "schema": "groups.qft.org/character-table/1", "versions": { "schema": "2.0.0", "dataset": "2026.11-workbench.4", "application": "6.0.0", "algorithm": "groups-workbench-4.0.0", "released": "2026-11-08" }, "id": "c3", "atlasId": "group:c-3", "name": "Cyclic group of order three", "symbol": "C₃", "latex": "C_3", "order": "3", "smallGroup": [ 3, 1 ], "field": "omega", "category": "Finite-dimensional ordinary complex representations", "coefficientField": "C", "characteristic": 0, "definition": "C3 = .", "classConvention": "The two nonidentity classes are the chosen generator r and its square.", "derivation": "All irreps are one-dimensional, with r acting by 1, omega, or omega squared. Powers are calculated modulo three.", "fieldDefinition": "ω = exp(2πi/3), ω² + ω + 1 = 0", "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" } ], "irreps": [ { "id": "1", "label": "1", "latex": "\\mathbf{1}", "labelTeX": "\\mathbf{1}", "dimension": "1", "reality": "Real", "indicator": 1, "conjugate": "1", "description": "Trivial representation", "characters": [ { "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" } ] }, { "id": "chi", "label": "χ", "latex": "\\chi", "labelTeX": "\\chi", "dimension": "1", "reality": "Complex", "indicator": 0, "conjugate": "chi2", "description": "The chosen generator acts by omega", "characters": [ { "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" } ] }, { "id": "chi2", "label": "χ²", "latex": "\\chi^2", "labelTeX": "\\chi^2", "dimension": "1", "reality": "Complex", "indicator": 0, "conjugate": "chi", "description": "The chosen generator acts by omega squared", "characters": [ { "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" } ] } ], "powerMaps": { "1": [ "1", "3a", "3b" ], "2": [ "1", "3b", "3a" ], "3": [ "1", "1", "1" ], "4": [ "1", "3a", "3b" ], "5": [ "1", "3b", "3a" ], "6": [ "1", "1", "1" ], "7": [ "1", "3a", "3b" ], "8": [ "1", "3b", "3a" ], "9": [ "1", "1", "1" ], "10": [ "1", "3a", "3b" ], "11": [ "1", "3b", "3a" ], "12": [ "1", "1", "1" ] }, "sources": [ { "id": "finite-groupnames-c3", "title": "Tim Dokchitser, GroupNames: C3", "url": "https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C3.html", "locator": "Character table, class sizes, group presentation, and representation type. Our explicit representatives and row labels fix any reordering.", "kind": "database", "checked": "2026-11-08", "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-etingof", "title": "Etingof et al., Introduction to Representation Theory", "url": "https://math.mit.edu/~etingof/reprbook.pdf", "locator": "Sections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicators", "kind": "lecture-notes", "checked": "2026-11-08", "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-characters", "title": "GAP Reference Manual, Chapter 72: Class Functions", "url": "https://gap-system.github.io/gap/doc/ref/chap72_mj.html", "locator": "Sections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powers", "kind": "documentation", "checked": "2026-11-08", "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-tables", "title": "GAP Reference Manual, Chapter 71: Character Tables", "url": "https://gap-system.github.io/gap/doc/ref/chap71_mj.html", "locator": "Section 71.12-5: second Frobenius–Schur indicator; class data and power maps", "kind": "documentation", "checked": "2026-11-08", "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", "url": "https://gap-system.github.io/gap/doc/ref/chap73_mj.html", "locator": "Section 73.1: power maps are extra class data and need not follow from the character matrix alone", "kind": "documentation", "checked": "2026-11-08", "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." } ], "conventionId": "finite-ordinary-complex-v1", "conventionVersion": "1.0.0", "convention": { "id": "finite-ordinary-complex-v1", "version": "1.0.0", "field": "Complex numbers; characteristic zero", "category": "Finite-dimensional ordinary complex representations", "innerProduct": "sum_C |C| chi(C) conjugate(psi(C)) / |G|", "dihedralNotation": "D8 is the symmetry group of the square, of order eight; GroupNames calls it D4.", "omega": "exp(2 pi i / 3)", "phi": "(1 + sqrt(5)) / 2", "representationLabels": "Labels and conjugacy-class orders are local conventions, recorded with each table. They must not be identified with upstream row numbers.", "scope": "Character decompositions do not specify Clebsch–Gordan coefficients, representation matrices, or a complete Lorentz-invariant operator basis." }, "algorithmVersion": "finite-characters-1.0.0", "dataVersion": "small-characters-1.0.0", "checks": { "group": "c3", "rows": 3, "classes": 3, "classSizes": true, "dimensionSquares": true, "rowOrthogonality": true, "columnOrthogonality": true, "frobeniusSchur": true, "conjugates": true, "powerMaps": true }, "independentExpertReview": "pending", "limits": { "maxDegree": 12, "maxInputMultiplicity": 12, "maxInputDimension": 60 } }