{
  "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/spin-10-weight-0-0-0-1-0/",
  "entity": {
    "id": "representation:spin-10-weight-0-0-0-1-0",
    "slug": "spin-10-weight-0-0-0-1-0",
    "name": "Conjugate chiral spinor of Spin(10)",
    "symbol": "D5 · (0,0,0,1,0)",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [
      "Spin(10)",
      "Conjugate chiral spinor",
      "16-dimensional",
      "Complex"
    ],
    "tags": [
      "Highest weight",
      "Compact",
      "Irreducible"
    ],
    "summary": "Highest weight (0,0,0,1,0) of the specified simply connected compact group: complex dimension 16, complex type, with exact dimension, Casimir, and center checks.",
    "definition": "The finite-dimensional continuous complex irreducible representation of Spin(10) with Dynkin labels (0,0,0,1,0) in the displayed Bourbaki/Sage node order.",
    "definitionEvidence": [
      {
        "source": "workbench-etingof-lie",
        "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
      },
      {
        "source": "workbench-sage-roots",
        "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
      },
      {
        "source": "sage",
        "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
      },
      {
        "source": "conrad-centers",
        "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
      },
      {
        "source": "workbench-adams-indicator",
        "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
      }
    ],
    "properties": [
      {
        "key": "dimension",
        "label": "Complex dimension",
        "value": "16",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ],
        "derivation": {
          "method": "Exact Weyl dimension product over positive coroots",
          "inputs": [
            "group:spin-10",
            "source:workbench-etingof-lie",
            "source:workbench-sage-roots",
            "source:sage",
            "source:conrad-centers",
            "source:workbench-adams-indicator"
          ]
        },
        "assumptions": []
      },
      {
        "key": "field",
        "label": "Coefficient field",
        "value": "ℂ; characteristic zero",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ],
        "assumptions": []
      },
      {
        "key": "casimir",
        "label": "Quadratic Casimir C₂",
        "value": "45/8",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ],
        "derivation": {
          "method": "(lambda,lambda+2rho)/2 with squared long-root length two",
          "inputs": [
            "group:spin-10",
            "source:workbench-etingof-lie",
            "source:workbench-sage-roots",
            "source:sage",
            "source:conrad-centers",
            "source:workbench-adams-indicator"
          ]
        },
        "assumptions": []
      },
      {
        "key": "index",
        "label": "Quadratic index T(R)",
        "value": "2",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ],
        "derivation": {
          "method": "T(R) dim(g) = C2(R) dim(R)",
          "inputs": [
            "group:spin-10",
            "source:workbench-etingof-lie",
            "source:workbench-sage-roots",
            "source:sage",
            "source:conrad-centers",
            "source:workbench-adams-indicator"
          ]
        },
        "assumptions": []
      },
      {
        "key": "reality",
        "label": "Reality type",
        "value": "Complex",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ],
        "assumptions": []
      },
      {
        "key": "conjugate",
        "label": "Conjugate highest weight",
        "value": "(0,0,0,0,1)",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ],
        "assumptions": []
      },
      {
        "key": "adjoint-descent",
        "label": "Descends to the adjoint global form",
        "value": "No",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ],
        "derivation": {
          "method": "Highest weight lies in the root lattice exactly when the full center acts trivially",
          "inputs": [
            "group:spin-10",
            "source:workbench-etingof-lie",
            "source:workbench-sage-roots",
            "source:sage",
            "source:conrad-centers",
            "source:workbench-adams-indicator"
          ]
        },
        "assumptions": []
      },
      {
        "key": "central-kernel-order",
        "label": "Order of central kernel",
        "value": "1",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ],
        "derivation": {
          "method": "Order of the center divided by the order of this central character",
          "inputs": [
            "group:spin-10",
            "source:workbench-etingof-lie",
            "source:workbench-sage-roots",
            "source:sage",
            "source:conrad-centers",
            "source:workbench-adams-indicator"
          ]
        },
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "representations",
        "title": "Highest weight and conventions",
        "paragraphs": [
          {
            "text": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.",
            "evidence": [
              {
                "source": "workbench-etingof-lie",
                "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
              },
              {
                "source": "workbench-sage-roots",
                "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
              },
              {
                "source": "sage",
                "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
              },
              {
                "source": "conrad-centers",
                "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
              },
              {
                "source": "workbench-adams-indicator",
                "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "global-form",
        "title": "Global form and center",
        "paragraphs": [
          {
            "text": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.",
            "evidence": [
              {
                "source": "workbench-etingof-lie",
                "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
              },
              {
                "source": "workbench-sage-roots",
                "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
              },
              {
                "source": "sage",
                "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
              },
              {
                "source": "conrad-centers",
                "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
              },
              {
                "source": "workbench-adams-indicator",
                "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
              }
            ],
            "assumptions": []
          }
        ]
      },
      {
        "id": "physics",
        "title": "Use in a calculation",
        "paragraphs": [
          {
            "text": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.",
            "evidence": [
              {
                "source": "workbench-etingof-lie",
                "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
              },
              {
                "source": "workbench-sage-roots",
                "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
              },
              {
                "source": "sage",
                "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
              },
              {
                "source": "conrad-centers",
                "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
              },
              {
                "source": "workbench-adams-indicator",
                "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "representation-of",
        "target": "group:spin-10",
        "label": "Representation domain",
        "evidence": [
          {
            "source": "workbench-etingof-lie",
            "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
          },
          {
            "source": "workbench-sage-roots",
            "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
          },
          {
            "source": "sage",
            "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
          },
          {
            "source": "conrad-centers",
            "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
          },
          {
            "source": "workbench-adams-indicator",
            "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
          }
        ]
      }
    ],
    "capabilities": [
      "lie-representations"
    ],
    "kind": "representation",
    "data": {
      "domain": "group:spin-10",
      "category": "finite-dimensional-continuous",
      "field": "C",
      "dimension": 16,
      "labels": "(0,0,0,1,0)",
      "displayTex": "D_{5}:\\quad V_{(0,0,0,1,0)}",
      "dynkinLabels": [
        0,
        0,
        0,
        1,
        0
      ],
      "conventionId": "lie-bourbaki-long-root-2-v1",
      "calculation": {
        "key": "D5",
        "type": "D",
        "rank": 5,
        "cartanMatrix": [
          [
            2,
            -1,
            0,
            0,
            0
          ],
          [
            -1,
            2,
            -1,
            0,
            0
          ],
          [
            0,
            -1,
            2,
            -1,
            -1
          ],
          [
            0,
            0,
            -1,
            2,
            0
          ],
          [
            0,
            0,
            -1,
            0,
            2
          ]
        ],
        "rootLengthsSquared": [
          "2",
          "2",
          "2",
          "2",
          "2"
        ],
        "positiveCorootCount": 20,
        "algebraDimension": 45,
        "simplyConnectedGroup": "Spin(10)",
        "center": "Z/4",
        "centerOrder": 4,
        "dualCoxeterNumber": 8,
        "adjointLabels": [
          0,
          1,
          0,
          0,
          0
        ],
        "exampleLabels": [
          0,
          0,
          0,
          0,
          1
        ],
        "numbering": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights.",
        "conventionId": "lie-bourbaki-long-root-2-v1",
        "cartanConvention": "A_ij = <alpha_j, alpha_i^vee>",
        "normalization": "Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2.",
        "labels": [
          0,
          0,
          0,
          1,
          0
        ],
        "dimension": "16",
        "casimir": {
          "numerator": "45",
          "denominator": "8",
          "text": "45/8",
          "latex": "\\frac{45}{8}"
        },
        "index": {
          "numerator": "2",
          "denominator": "1",
          "text": "2",
          "latex": "2"
        },
        "conjugate": [
          0,
          0,
          0,
          0,
          1
        ],
        "selfDual": false,
        "reality": "Complex",
        "frobeniusSchurIndicator": 0,
        "adjointDescent": false,
        "centerCharacterOrder": "4",
        "centerKernelOrder": "1",
        "centerCharge": null,
        "centerChargeModulus": null,
        "rootBasisCoordinates": [
          {
            "numerator": "1",
            "denominator": "2",
            "text": "1/2",
            "latex": "\\frac{1}{2}"
          },
          {
            "numerator": "1",
            "denominator": "1",
            "text": "1",
            "latex": "1"
          },
          {
            "numerator": "3",
            "denominator": "2",
            "text": "3/2",
            "latex": "\\frac{3}{2}"
          },
          {
            "numerator": "5",
            "denominator": "4",
            "text": "5/4",
            "latex": "\\frac{5}{4}"
          },
          {
            "numerator": "3",
            "denominator": "4",
            "text": "3/4",
            "latex": "\\frac{3}{4}"
          }
        ],
        "positiveCoroots": [
          [
            0,
            0,
            0,
            0,
            1
          ],
          [
            0,
            0,
            0,
            1,
            0
          ],
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            0,
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          ],
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          ],
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          ],
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          ],
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          ],
          [
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          ],
          [
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          ],
          [
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            0
          ],
          [
            0,
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            1,
            1,
            1
          ],
          [
            1,
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            1
          ],
          [
            1,
            1,
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            1,
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          ],
          [
            0,
            1,
            2,
            1,
            1
          ],
          [
            1,
            1,
            1,
            1,
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          ],
          [
            1,
            1,
            2,
            1,
            1
          ],
          [
            1,
            2,
            2,
            1,
            1
          ]
        ],
        "factors": [
          {
            "coroot": [
              0,
              0,
              0,
              0,
              1
            ],
            "numerator": "1",
            "denominator": "1"
          },
          {
            "coroot": [
              0,
              0,
              0,
              1,
              0
            ],
            "numerator": "2",
            "denominator": "1"
          },
          {
            "coroot": [
              0,
              0,
              1,
              0,
              0
            ],
            "numerator": "1",
            "denominator": "1"
          },
          {
            "coroot": [
              0,
              1,
              0,
              0,
              0
            ],
            "numerator": "1",
            "denominator": "1"
          },
          {
            "coroot": [
              1,
              0,
              0,
              0,
              0
            ],
            "numerator": "1",
            "denominator": "1"
          },
          {
            "coroot": [
              0,
              0,
              1,
              0,
              1
            ],
            "numerator": "2",
            "denominator": "2"
          },
          {
            "coroot": [
              0,
              0,
              1,
              1,
              0
            ],
            "numerator": "3",
            "denominator": "2"
          },
          {
            "coroot": [
              0,
              1,
              1,
              0,
              0
            ],
            "numerator": "2",
            "denominator": "2"
          },
          {
            "coroot": [
              1,
              1,
              0,
              0,
              0
            ],
            "numerator": "2",
            "denominator": "2"
          },
          {
            "coroot": [
              0,
              0,
              1,
              1,
              1
            ],
            "numerator": "4",
            "denominator": "3"
          },
          {
            "coroot": [
              0,
              1,
              1,
              0,
              1
            ],
            "numerator": "3",
            "denominator": "3"
          },
          {
            "coroot": [
              0,
              1,
              1,
              1,
              0
            ],
            "numerator": "4",
            "denominator": "3"
          },
          {
            "coroot": [
              1,
              1,
              1,
              0,
              0
            ],
            "numerator": "3",
            "denominator": "3"
          },
          {
            "coroot": [
              0,
              1,
              1,
              1,
              1
            ],
            "numerator": "5",
            "denominator": "4"
          },
          {
            "coroot": [
              1,
              1,
              1,
              0,
              1
            ],
            "numerator": "4",
            "denominator": "4"
          },
          {
            "coroot": [
              1,
              1,
              1,
              1,
              0
            ],
            "numerator": "5",
            "denominator": "4"
          },
          {
            "coroot": [
              0,
              1,
              2,
              1,
              1
            ],
            "numerator": "6",
            "denominator": "5"
          },
          {
            "coroot": [
              1,
              1,
              1,
              1,
              1
            ],
            "numerator": "6",
            "denominator": "5"
          },
          {
            "coroot": [
              1,
              1,
              2,
              1,
              1
            ],
            "numerator": "7",
            "denominator": "6"
          },
          {
            "coroot": [
              1,
              2,
              2,
              1,
              1
            ],
            "numerator": "8",
            "denominator": "7"
          }
        ],
        "algorithmVersion": "lie-representations-1.0.0",
        "category": "Finite-dimensional continuous complex irreducible representations of the compact simply connected group",
        "method": "Exact Weyl product over positive coroots; rational weight-metric Casimir; root-lattice descent; central-character Frobenius–Schur indicator.",
        "checks": {
          "positiveCorootCount": true,
          "dimensionIntegral": true,
          "weightGramSymmetric": true,
          "centralCharacterOrderDividesCenter": true
        },
        "sources": [
          {
            "id": "workbench-etingof-lie",
            "title": "Lie Groups and Lie Algebras I",
            "authors": [
              "Pavel Etingof"
            ],
            "url": "https://math.mit.edu/~etingof/lnlg.pdf",
            "kind": "lecture-notes",
            "locatorHint": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations.",
            "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
          },
          {
            "id": "workbench-sage-roots",
            "title": "SageMath: Cartan matrices",
            "authors": [
              "SageMath developers"
            ],
            "url": "https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/cartan_matrix.html",
            "kind": "documentation",
            "locatorHint": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08.",
            "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
          },
          {
            "id": "workbench-sage-weyl",
            "title": "SageMath: The Weyl Character Ring",
            "authors": [
              "Daniel Bump",
              "SageMath developers"
            ],
            "url": "https://doc.sagemath.org/html/en/thematic_tutorials/lie/weyl_character_ring.html",
            "kind": "documentation",
            "locatorHint": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL.",
            "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
          },
          {
            "id": "workbench-conrad-centers",
            "title": "Centers of Simply Connected Semisimple Groups",
            "authors": [
              "Brian Conrad"
            ],
            "url": "https://math.stanford.edu/~conrad/210CPage/handouts/centers.pdf",
            "kind": "lecture-notes",
            "locatorHint": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated.",
            "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
          },
          {
            "id": "workbench-adams-indicator",
            "title": "The Real Chevalley Involution",
            "authors": [
              "Jeffrey Adams"
            ],
            "url": "https://www.math.umd.edu/~jda/preprints/realchevalley.pdf",
            "kind": "paper",
            "locatorHint": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep.",
            "licenseNote": "Linked primary source. Original algorithms and explanatory prose; no upstream software or document text is redistributed."
          }
        ]
      },
      "capabilityPresets": {
        "lie-representations": {
          "type": "D",
          "rank": "5",
          "labels": "0,0,0,1,0"
        }
      }
    }
  },
  "claims": [
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#definition",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "definition",
      "label": "Definition",
      "value": "The finite-dimensional continuous complex irreducible representation of Spin(10) with Dynkin labels (0,0,0,1,0) in the displayed Bourbaki/Sage node order.",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#dimension",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "dimension",
      "label": "Complex dimension",
      "value": "16",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "derivation": {
        "method": "Exact Weyl dimension product over positive coroots",
        "inputs": [
          "group:spin-10",
          "source:workbench-etingof-lie",
          "source:workbench-sage-roots",
          "source:sage",
          "source:conrad-centers",
          "source:workbench-adams-indicator"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#field",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "field",
      "label": "Coefficient field",
      "value": "ℂ; characteristic zero",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#casimir",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "casimir",
      "label": "Quadratic Casimir C₂",
      "value": "45/8",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "derivation": {
        "method": "(lambda,lambda+2rho)/2 with squared long-root length two",
        "inputs": [
          "group:spin-10",
          "source:workbench-etingof-lie",
          "source:workbench-sage-roots",
          "source:sage",
          "source:conrad-centers",
          "source:workbench-adams-indicator"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#index",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "index",
      "label": "Quadratic index T(R)",
      "value": "2",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "derivation": {
        "method": "T(R) dim(g) = C2(R) dim(R)",
        "inputs": [
          "group:spin-10",
          "source:workbench-etingof-lie",
          "source:workbench-sage-roots",
          "source:sage",
          "source:conrad-centers",
          "source:workbench-adams-indicator"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#reality",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "reality",
      "label": "Reality type",
      "value": "Complex",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#conjugate",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "conjugate",
      "label": "Conjugate highest weight",
      "value": "(0,0,0,0,1)",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#adjoint-descent",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "adjoint-descent",
      "label": "Descends to the adjoint global form",
      "value": "No",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "derivation": {
        "method": "Highest weight lies in the root lattice exactly when the full center acts trivially",
        "inputs": [
          "group:spin-10",
          "source:workbench-etingof-lie",
          "source:workbench-sage-roots",
          "source:sage",
          "source:conrad-centers",
          "source:workbench-adams-indicator"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#central-kernel-order",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "central-kernel-order",
      "label": "Order of central kernel",
      "value": "1",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "derivation": {
        "method": "Order of the center divided by the order of this central character",
        "inputs": [
          "group:spin-10",
          "source:workbench-etingof-lie",
          "source:workbench-sage-roots",
          "source:sage",
          "source:conrad-centers",
          "source:workbench-adams-indicator"
        ]
      },
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#representations-1",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "representations-1",
      "label": "Highest weight and conventions",
      "value": "The chain 1,…,r−2 branches to nodes r−1 and r. Those two end nodes carry the half-spin highest weights. Long roots have squared length 2; C2(lambda) = (lambda,lambda+2 rho)/2. Complex dimension counts the underlying complex vector space; reality type is an additional invariant.",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#global-form-1",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "global-form-1",
      "label": "Global form and center",
      "value": "The domain is the stated simply connected compact group. The test for descent to its adjoint global form gives no. This divides out the entire center; a different central quotient must be checked against its particular embedded kernel.",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#physics-1",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "physics-1",
      "label": "Use in a calculation",
      "value": "Choose the exact domain and highest weight when entering a multiplet. A dimension label can be ambiguous, especially for conjugate spinors or representations related by triality. A tensor multiplicity or Casimir does not determine a basis of matrices or intertwiners.",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "representation:spin-10-weight-0-0-0-1-0#relation-1",
      "subject": "representation:spin-10-weight-0-0-0-1-0",
      "key": "relation-1",
      "label": "Representation domain",
      "value": "group:spin-10",
      "evidence": [
        {
          "source": "workbench-etingof-lie",
          "locator": "Sections 25.3–25.4, 28.1–28.4: characters, Casimir eigenvalues, Weyl dimensions and type-A representations."
        },
        {
          "source": "workbench-sage-roots",
          "locator": "Finite Cartan matrices and root-system conventions; documentation checked 2026-11-08."
        },
        {
          "source": "sage",
          "locator": "Coroot notation; tensor products; symmetric and exterior powers; Weyl dimension formula; SL versus GL."
        },
        {
          "source": "conrad-centers",
          "locator": "Pages 1–4: the center is dual to P/Q, with classical and exceptional types tabulated."
        },
        {
          "source": "workbench-adams-indicator",
          "locator": "Lemma 5.3, equation (5.4), printed pages 16–17: the indicator of a self-dual finite-dimensional irrep."
        }
      ],
      "assumptions": [],
      "relationship": "representation-of",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "sources": [
    {
      "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": "conrad-centers",
      "title": "Math 210C. Centers of simply connected semisimple groups",
      "authors": [
        "Brian Conrad"
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      "title": "The Real Chevalley Involution",
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