{
  "schema": "groups.qft.org/entity/2",
  "versions": {
    "schema": "2.0.0",
    "dataset": "2026.11-workbench.4",
    "application": "6.0.0",
    "algorithm": "groups-workbench-4.0.0",
    "released": "2026-11-08"
  },
  "canonicalURL": "https://groups.qft.org/atlas/representations/q-8-b/",
  "entity": {
    "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"
        }
      }
    }
  },
  "claims": [
    {
      "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"
    }
  ],
  "sources": [
    {
      "id": "indicators",
      "title": "GAP · Character tables and Frobenius–Schur indicators",
      "authors": [
        "The GAP Group"
      ],
      "url": "https://gap-system.github.io/gap/doc/ref/chap71_mj.html",
      "kind": "documentation",
      "locatorHint": "Irr; DegreesOfIrreducibleCharacters; section 71.12-5 Indicator",
      "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": "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": "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."
    }
  ],
  "review": {
    "status": "pending-expert-review",
    "label": "Expert review pending",
    "currentReviews": 0,
    "staleReviews": 0,
    "fingerprint": "59a1f21c69a290a1e2f340b964b8562b903bbe1024a8545ead8fb3f92c8f0076",
    "reviewers": []
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}