Lie theory · Map
The diagonal-Z6 Standard Model group inside SU(5)
The injective group map induced by diag(z⁻² A,z³ B) in the Q=6Y convention.
Reference coverageEmbeddingBlock subgroup
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)).
Definition sourcesThe group map
Reference properties
What the embedding establishes
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. [sources]
Related objects and maps
- G_SM^(6)The specified diagonal quotientStandard Model global group with diagonal Z6 quotient
- Parent compact group
- SU(3) × SU(2) × U(1)_X → SU(5)Computable block map in the X conventionColor-first SU(5) block homomorphism
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
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)).
- 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
map:standard-model-z6-into-su5#definitionKernel
Trivial
- 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
map:standard-model-z6-into-su5#kernelInjective
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
map:standard-model-z6-into-su5#injectiveImage
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
map:standard-model-z6-into-su5#imageWhat the embedding establishes
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.
- 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
map:standard-model-z6-into-su5#physics-1The specified diagonal quotient
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
map:standard-model-z6-into-su5#relation-1Parent compact group
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
map:standard-model-z6-into-su5#relation-2Computable block map in the X convention
map:su-3-su-2-u1-to-su-5-block
- 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
map:standard-model-z6-into-su5#relation-3