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  "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/maps/su-4-to-so-6/",
  "entity": {
    "id": "map:su-4-to-so-6",
    "slug": "su-4-to-so-6",
    "name": "Exterior-square realization of the Spin(6) cover",
    "symbol": "SU(4) → SO(6)",
    "category": "lie",
    "level": "reference",
    "aliases": [],
    "searchTerms": [],
    "tags": [
      "Covering or central quotient"
    ],
    "summary": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form. Kernel: {±I} ≅ C₂.",
    "definition": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.",
    "definitionEvidence": [
      {
        "source": "garrett-isogenies",
        "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
      }
    ],
    "properties": [
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        "key": "domain",
        "label": "Domain",
        "value": "su-4",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ],
        "assumptions": []
      },
      {
        "key": "codomain",
        "label": "Codomain",
        "value": "so-6",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ],
        "assumptions": []
      },
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        "key": "formula",
        "label": "Specified homomorphism",
        "value": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ],
        "assumptions": []
      },
      {
        "key": "kernel",
        "label": "Kernel",
        "value": "{±I} ≅ C₂",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ],
        "assumptions": []
      },
      {
        "key": "image",
        "label": "Image",
        "value": "SO(6)",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ],
        "assumptions": []
      },
      {
        "key": "injective",
        "label": "Injective",
        "value": "No",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ],
        "assumptions": []
      },
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        "key": "surjective",
        "label": "Surjective",
        "value": "Yes",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ],
        "assumptions": []
      },
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        "key": "covering-degree",
        "label": "Covering degree",
        "value": "2",
        "evidence": [
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            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "meaning",
        "title": "Global and representation consequences",
        "paragraphs": [
          {
            "text": "This identifies SU(4) with Spin(6). The central element iI acts by −I on the vector 6, while −I acts trivially; quotienting by the full C₄ center would give PSO(6).",
            "evidence": [
              {
                "source": "garrett-isogenies",
                "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
              },
              {
                "source": "etingof-global-forms",
                "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-domain",
        "target": "group:su-4",
        "label": "Domain group",
        "evidence": [
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            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
        ]
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      {
        "type": "has-codomain",
        "target": "group:so-6",
        "label": "Codomain group",
        "evidence": [
          {
            "source": "garrett-isogenies",
            "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
          }
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    "capabilities": [],
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    "data": {
      "domain": "group:su-4",
      "codomain": "group:so-6",
      "kernel": "{±I} ≅ C₂",
      "image": "SO(6)",
      "injective": false,
      "surjective": true
    }
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      "key": "definition",
      "label": "Definition",
      "value": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
        }
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          "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
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      "value": "so-6",
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    },
    {
      "id": "map:su-4-to-so-6#formula",
      "subject": "map:su-4-to-so-6",
      "key": "formula",
      "label": "Specified homomorphism",
      "value": "A ↦ Λ²A restricted to the invariant real six-dimensional subspace of Λ²C⁴. The wedge pairing and invariant Hermitian structure determine this real form.",
      "evidence": [
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          "source": "garrett-isogenies",
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      "assumptions": [],
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      "id": "map:su-4-to-so-6#kernel",
      "subject": "map:su-4-to-so-6",
      "key": "kernel",
      "label": "Kernel",
      "value": "{±I} ≅ C₂",
      "evidence": [
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          "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
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      "key": "image",
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        }
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      "revision": "2026.11-workbench.4"
    },
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      "id": "map:su-4-to-so-6#injective",
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      "key": "injective",
      "label": "Injective",
      "value": "No",
      "evidence": [
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          "source": "garrett-isogenies",
          "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
        }
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      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-4-to-so-6#surjective",
      "subject": "map:su-4-to-so-6",
      "key": "surjective",
      "label": "Surjective",
      "value": "Yes",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
        }
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    {
      "id": "map:su-4-to-so-6#covering-degree",
      "subject": "map:su-4-to-so-6",
      "key": "covering-degree",
      "label": "Covering degree",
      "value": "2",
      "evidence": [
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          "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
        }
      ],
      "assumptions": [],
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      "id": "map:su-4-to-so-6#meaning-1",
      "subject": "map:su-4-to-so-6",
      "key": "meaning-1",
      "label": "Global and representation consequences",
      "value": "This identifies SU(4) with Spin(6). The central element iI acts by −I on the vector 6, while −I acts trivially; quotienting by the full C₄ center would give PSO(6).",
      "evidence": [
        {
          "source": "garrett-isogenies",
          "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
        },
        {
          "source": "etingof-global-forms",
          "locator": "Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients"
        }
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      "label": "Domain group",
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          "locator": "Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴"
        }
      ],
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      "relationship": "has-codomain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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  ],
  "sources": [
    {
      "id": "etingof-global-forms",
      "title": "Lecture 17: Classification of Connected Compact and Complex Reductive Groups",
      "authors": [
        "Pavel Etingof"
      ],
      "url": "https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/mit18_755_s24_lec17.pdf",
      "kind": "lecture-notes",
      "locatorHint": "Sections 43.1–43.2, printed pages 230–235.",
      "licenseNote": "Linked MIT OpenCourseWare notes; global-form and representation criteria are restated."
    },
    {
      "id": "garrett-isogenies",
      "title": "Sporadic isogenies to orthogonal groups",
      "authors": [
        "Paul Garrett"
      ],
      "url": "https://ncatlab.org/nlab/files/Garrett-SporadicIsogenies.pdf",
      "kind": "lecture-notes",
      "locatorHint": "May 7, 2015 notes, Sections 2.1–2.11. The linked file is an accessible mirror of the author’s notes.",
      "licenseNote": "Author’s lecture notes linked at an accessible mirror. No redistribution license is inferred; matrix constructions are explained independently."
    }
  ],
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