$SU(3)\times SU(2)\times U(1)_X\longrightarrow SU(5)$Color-first SU(5) block homomorphism

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)).

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

Z6 with the explicitly stated generator

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

S(U(3) × U(2))

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

No

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

No

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

Y=−X/6; integer Q=6Y=−X

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

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.

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

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.

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

group:standard-model-z1

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

group:su-5

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

group:standard-model-z6

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source