{
  "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/su-3-su-2-u1-to-su-5-block/",
  "entity": {
    "id": "map:su-3-su-2-u1-to-su-5-block",
    "slug": "su-3-su-2-u1-to-su-5-block",
    "name": "Color-first SU(5) block homomorphism",
    "symbol": "SU(3) × SU(2) × U(1)_X → SU(5)",
    "category": "lie",
    "level": "executable",
    "aliases": [],
    "searchTerms": [
      "SU5",
      "hypercharge",
      "branching",
      "block subgroup"
    ],
    "tags": [
      "Homomorphism",
      "Finite kernel",
      "Charge refined"
    ],
    "summary": "The block map diag(z² A,z⁻³ B), with primitive charge X and kernel Z6. Physical hypercharge is Y=−X/6.",
    "definition": "The homomorphism ι(A,B,z)=diag(z² A,z⁻³ B) from SU(3) × SU(2) × U(1) to SU(5), with the color block first. Its kernel is generated by (exp(−2πi/3) I₃,−I₂,exp(2πi/6)) and its image is 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": "Z6 with the explicitly stated generator",
        "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": []
      },
      {
        "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 onto SU(5)",
        "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": "normalization",
        "label": "Physical hypercharge",
        "value": "Y=−X/6; integer Q=6Y=−X",
        "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": "branching",
        "title": "Keep the charge",
        "paragraphs": [
          {
            "text": "The same nonabelian multiplet can occur at different U(1) charges. The complete output is labeled by both SU highest weights and X. The kernel condition is −2t3+3t2+X=0 mod 6. The tool separately displays Y=−X/6 and transfers Q=−X when opening the product-group calculation.",
            "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": []
          }
        ]
      },
      {
        "id": "global-form",
        "title": "From homomorphism to embedding",
        "paragraphs": [
          {
            "text": "The map is not injective on the compact product. Quotienting by its actual kernel gives the block subgroup. Inverting the U(1) coordinate changes from the X convention to Q=6Y and identifies this quotient with the G_SM^(6) record.",
            "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": "Underlying compact product, with the X coordinate stated here",
        "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 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": "image-isomorphic-to",
        "target": "group:standard-model-z6",
        "label": "Image after converting X to Q=−X",
        "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": [
      "charged-branching"
    ],
    "kind": "map",
    "data": {
      "domain": "group:standard-model-z1",
      "codomain": "group:su-5",
      "kernel": "⟨(ω₃⁻¹,−1,exp(2πi/6))⟩ ≅ Z6",
      "image": "S(U(3) × U(2))",
      "injective": false,
      "surjective": false,
      "displayTex": "SU(3)\\times SU(2)\\times U(1)_X\\longrightarrow SU(5)",
      "conventionId": "su-n-block-u1-v1",
      "capabilityPresets": {
        "charged-branching": {
          "n": "5",
          "p": "3",
          "labels": "0,1,0,0"
        }
      }
    }
  },
  "claims": [
    {
      "id": "map:su-3-su-2-u1-to-su-5-block#definition",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "definition",
      "label": "Definition",
      "value": "The homomorphism ι(A,B,z)=diag(z² A,z⁻³ B) from SU(3) × SU(2) × U(1) to SU(5), with the color block first. Its kernel is generated by (exp(−2πi/3) I₃,−I₂,exp(2πi/6)) and its image is 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:su-3-su-2-u1-to-su-5-block#kernel",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "kernel",
      "label": "Kernel",
      "value": "Z6 with the explicitly stated generator",
      "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:su-3-su-2-u1-to-su-5-block#image",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "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:su-3-su-2-u1-to-su-5-block#injective",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "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:su-3-su-2-u1-to-su-5-block#surjective",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "surjective",
      "label": "Surjective onto SU(5)",
      "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:su-3-su-2-u1-to-su-5-block#normalization",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "normalization",
      "label": "Physical hypercharge",
      "value": "Y=−X/6; integer Q=6Y=−X",
      "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:su-3-su-2-u1-to-su-5-block#branching-1",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "branching-1",
      "label": "Keep the charge",
      "value": "The same nonabelian multiplet can occur at different U(1) charges. The complete output is labeled by both SU highest weights and X. The kernel condition is −2t3+3t2+X=0 mod 6. The tool separately displays Y=−X/6 and transfers Q=−X when opening the product-group calculation.",
      "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:su-3-su-2-u1-to-su-5-block#global-form-1",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "global-form-1",
      "label": "From homomorphism to embedding",
      "value": "The map is not injective on the compact product. Quotienting by its actual kernel gives the block subgroup. Inverting the U(1) coordinate changes from the X convention to Q=6Y and identifies this quotient with the G_SM^(6) record.",
      "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:su-3-su-2-u1-to-su-5-block#relation-1",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "relation-1",
      "label": "Underlying compact product, with the X coordinate stated here",
      "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",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    },
    {
      "id": "map:su-3-su-2-u1-to-su-5-block#relation-2",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "relation-2",
      "label": "Parent group",
      "value": "group:su-5",
      "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"
    },
    {
      "id": "map:su-3-su-2-u1-to-su-5-block#relation-3",
      "subject": "map:su-3-su-2-u1-to-su-5-block",
      "key": "relation-3",
      "label": "Image after converting X to Q=−X",
      "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"
        }
      ],
      "assumptions": [],
      "relationship": "image-isomorphic-to",
      "reviewStatus": "pending-expert-review",
      "revision": "2026.11-workbench.4"
    }
  ],
  "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": "8ce75aff7d416a356a175806ffdfa15ee1e41f38a1ea398c9a0663bef30caa4a",
    "reviewers": []
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}