{
  "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-cover-to-z3/",
  "entity": {
    "id": "map:standard-model-cover-to-z3",
    "slug": "standard-model-cover-to-z3",
    "name": "Standard Model quotient map of order 3",
    "symbol": "G_SM^(1) → G_SM^(3)",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [
      "Standard Model",
      "quotient",
      "cover",
      "hypercharge"
    ],
    "tags": [
      "Central quotient",
      "Surjective"
    ],
    "summary": "The explicit quotient by the subgroup generated by γ^2, with γ=(ω₃,−1,exp(2πi/6)) and integer Q=6Y.",
    "definition": "The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_3=⟨γ^2⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 3.",
    "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": "Γ_3 = ⟨γ^2⟩ ≅ Z_3",
        "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": "No",
        "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": "surjective",
        "label": "Surjective",
        "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": []
      }
    ],
    "sections": [
      {
        "id": "matter",
        "title": "Matter descent",
        "paragraphs": [
          {
            "text": "A representation of the compact product descends exactly when every generator of this embedded kernel acts trivially. Test the color center, weak center, and abelian phase together; the same abstract cyclic group embedded in one factor can give a different answer.",
            "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": []
          }
        ]
      }
    ],
    "relationships": [
      {
        "type": "has-domain",
        "target": "group:standard-model-z1",
        "label": "Compact covering product",
        "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:standard-model-z3",
        "label": "Chosen global form",
        "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": [
      "product-gauge-groups"
    ],
    "kind": "map",
    "data": {
      "domain": "group:standard-model-z1",
      "codomain": "group:standard-model-z3",
      "kernel": "Γ_3 = ⟨γ^2⟩",
      "image": "G_SM^(3)",
      "injective": false,
      "surjective": true,
      "displayTex": "G_{\\mathrm{SM}}^{(1)}\\longrightarrow G_{\\mathrm{SM}}^{(3)}",
      "capabilityPresets": {
        "product-gauge-groups": {
          "title": "Standard Model matter with quotient order 3",
          "ranks": "3,2",
          "matter": "Q: 1,0 | 1 | 1\nu_c: 0,1 | 0 | -4\nd_c: 0,1 | 0 | 2\nL: 0,0 | 1 | -3\ne_c: 0,0 | 0 | 6\nH: 0,0 | 1 | 3",
          "generators": "2,0 | 1/3"
        }
      }
    }
  },
  "claims": [
    {
      "id": "map:standard-model-cover-to-z3#definition",
      "subject": "map:standard-model-cover-to-z3",
      "key": "definition",
      "label": "Definition",
      "value": "The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_3=⟨γ^2⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 3.",
      "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-cover-to-z3#kernel",
      "subject": "map:standard-model-cover-to-z3",
      "key": "kernel",
      "label": "Kernel",
      "value": "Γ_3 = ⟨γ^2⟩ ≅ Z_3",
      "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-cover-to-z3#injective",
      "subject": "map:standard-model-cover-to-z3",
      "key": "injective",
      "label": "Injective",
      "value": "No",
      "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-cover-to-z3#surjective",
      "subject": "map:standard-model-cover-to-z3",
      "key": "surjective",
      "label": "Surjective",
      "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-cover-to-z3#matter-1",
      "subject": "map:standard-model-cover-to-z3",
      "key": "matter-1",
      "label": "Matter descent",
      "value": "A representation of the compact product descends exactly when every generator of this embedded kernel acts trivially. Test the color center, weak center, and abelian phase together; the same abstract cyclic group embedded in one factor can give a different answer.",
      "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-cover-to-z3#relation-1",
      "subject": "map:standard-model-cover-to-z3",
      "key": "relation-1",
      "label": "Compact covering product",
      "value": "group:standard-model-z1",
      "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": [],
      "relationship": "has-domain",
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      "revision": "2026.11-workbench.4"
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      "id": "map:standard-model-cover-to-z3#relation-2",
      "subject": "map:standard-model-cover-to-z3",
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      "label": "Chosen global form",
      "value": "group:standard-model-z3",
      "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": [],
      "relationship": "has-codomain",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
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  ],
  "sources": [
    {
      "id": "tong",
      "title": "Tong · Line operators in the Standard Model",
      "authors": [
        "David Tong"
      ],
      "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"
      ],
      "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."
    }
  ],
  "review": {
    "status": "pending-expert-review",
    "label": "Expert review pending",
    "currentReviews": 0,
    "staleReviews": 0,
    "fingerprint": "68b832d65f92402d41459817dc5f4efaf9158ac1c98554b1542d96fffa325fee",
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}