{
  "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/maps/standard-model-z6-into-su5/",
  "entity": {
    "id": "map:standard-model-z6-into-su5",
    "slug": "standard-model-z6-into-su5",
    "name": "The diagonal-Z6 Standard Model group inside SU(5)",
    "symbol": "G_SM^(6) ↪ SU(5)",
    "category": "lie",
    "level": "reference",
    "aliases": [],
    "searchTerms": [
      "Standard Model",
      "SU5",
      "embedding",
      "hypercharge"
    ],
    "tags": [
      "Embedding",
      "Block subgroup"
    ],
    "summary": "The injective group map induced by diag(z⁻² A,z³ B) in the Q=6Y convention.",
    "definition": "The map [(A,B,z)] ↦ diag(z⁻² A,z³ B) from G_SM^(6) to SU(5). The numerator map has precisely the diagonal Γ6 kernel, so it descends to an injective homomorphism with image S(U(3) × U(2)).",
    "definitionEvidence": [
      {
        "source": "block-baez-huerta",
        "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
      },
      {
        "source": "tong",
        "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
      },
      {
        "source": "conrad-pi1-lie",
        "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
      }
    ],
    "properties": [
      {
        "key": "kernel",
        "label": "Kernel",
        "value": "Trivial",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      },
      {
        "key": "injective",
        "label": "Injective",
        "value": "Yes",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      },
      {
        "key": "image",
        "label": "Image",
        "value": "S(U(3) × U(2))",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ],
        "assumptions": []
      }
    ],
    "sections": [
      {
        "id": "physics",
        "title": "What the embedding establishes",
        "paragraphs": [
          {
            "text": "The map provides the global group and charge conventions for SU(5) branching. It does not establish that the observed Standard Model is unified in SU(5), or select a Higgs potential, symmetry-breaking vacuum, or ultraviolet dynamics.",
            "evidence": [
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                "source": "block-baez-huerta",
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              },
              {
                "source": "tong",
                "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
              },
              {
                "source": "conrad-pi1-lie",
                "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
              }
            ],
            "assumptions": []
          }
        ]
      }
    ],
    "relationships": [
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        "type": "has-domain",
        "target": "group:standard-model-z6",
        "label": "The specified diagonal quotient",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ]
      },
      {
        "type": "has-codomain",
        "target": "group:su-5",
        "label": "Parent compact group",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ]
      },
      {
        "type": "related-map",
        "target": "map:su-3-su-2-u1-to-su-5-block",
        "label": "Computable block map in the X convention",
        "evidence": [
          {
            "source": "block-baez-huerta",
            "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
          },
          {
            "source": "tong",
            "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
          },
          {
            "source": "conrad-pi1-lie",
            "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
          }
        ]
      }
    ],
    "capabilities": [],
    "kind": "map",
    "data": {
      "domain": "group:standard-model-z6",
      "codomain": "group:su-5",
      "kernel": "1",
      "image": "S(U(3) × U(2))",
      "injective": true,
      "surjective": false,
      "displayTex": "G_{\\mathrm{SM}}^{(6)}\\hookrightarrow SU(5)",
      "conventionId": "product-su-u1-center-v1"
    }
  },
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      "subject": "map:standard-model-z6-into-su5",
      "key": "definition",
      "label": "Definition",
      "value": "The map [(A,B,z)] ↦ diag(z⁻² A,z³ B) from G_SM^(6) to SU(5). The numerator map has precisely the diagonal Γ6 kernel, so it descends to an injective homomorphism with image S(U(3) × U(2)).",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:standard-model-z6-into-su5#kernel",
      "subject": "map:standard-model-z6-into-su5",
      "key": "kernel",
      "label": "Kernel",
      "value": "Trivial",
      "evidence": [
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          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
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      "revision": "2026.11-workbench.4"
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      "id": "map:standard-model-z6-into-su5#injective",
      "subject": "map:standard-model-z6-into-su5",
      "key": "injective",
      "label": "Injective",
      "value": "Yes",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:standard-model-z6-into-su5#image",
      "subject": "map:standard-model-z6-into-su5",
      "key": "image",
      "label": "Image",
      "value": "S(U(3) × U(2))",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:standard-model-z6-into-su5#physics-1",
      "subject": "map:standard-model-z6-into-su5",
      "key": "physics-1",
      "label": "What the embedding establishes",
      "value": "The map provides the global group and charge conventions for SU(5) branching. It does not establish that the observed Standard Model is unified in SU(5), or select a Higgs potential, symmetry-breaking vacuum, or ultraviolet dynamics.",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
        }
      ],
      "assumptions": [],
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:standard-model-z6-into-su5#relation-1",
      "subject": "map:standard-model-z6-into-su5",
      "key": "relation-1",
      "label": "The specified diagonal quotient",
      "value": "group:standard-model-z6",
      "evidence": [
        {
          "source": "block-baez-huerta",
          "locator": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly."
        },
        {
          "source": "tong",
          "locator": "§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices"
        },
        {
          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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      "label": "Parent compact group",
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          "source": "block-baez-huerta",
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        {
          "source": "tong",
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          "source": "conrad-pi1-lie",
          "locator": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup"
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          "source": "tong",
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  "sources": [
    {
      "id": "tong",
      "title": "Tong · Line operators in the Standard Model",
      "authors": [
        "David Tong"
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      "url": "https://arxiv.org/abs/1705.01853",
      "kind": "paper",
      "locatorHint": "Introduction and Section 3: the Z6 quotient, integer-normalized hypercharge, and the additional choice of line operators.",
      "licenseNote": "Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted."
    },
    {
      "id": "block-baez-huerta",
      "title": "The Algebra of Grand Unified Theories",
      "authors": [
        "John C. Baez",
        "John Huerta"
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      "url": "https://arxiv.org/abs/0904.1556",
      "kind": "paper",
      "locatorHint": "Section 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.",
      "licenseNote": "Linked primary source supporting original derivations and calculations. No upstream text, software, or representation dataset is redistributed."
    },
    {
      "id": "conrad-pi1-lie",
      "title": "Brian Conrad, Fundamental Groups of Lie Groups",
      "authors": [
        "Brian Conrad"
      ],
      "url": "https://math.stanford.edu/~conrad/210CPage/handouts/pi1stuff.pdf",
      "kind": "lecture-notes",
      "locatorHint": "pp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup",
      "licenseNote": "Linked primary source; the product-group specialization is derived in the local method page."
    }
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