$G_{\mathrm{SM}}^{(1)}\longrightarrow G_{\mathrm{SM}}^{(2)}$Standard Model quotient map of order 2

The natural projection from SU(3) × SU(2) × U(1) onto its quotient by Γ_2=⟨γ^3⟩, with γ=(exp(2πi/3) I₃,−I₂,exp(2πi/6)). Its kernel is the specified cyclic central subgroup of order 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

Γ_2 = ⟨γ^3⟩ ≅ Z_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

Yes

• 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

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.

• 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:standard-model-z2

• 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