Delta(27), the finite Heisenberg group
The nonabelian Heisenberg group over F₃, with nine singlets and two conjugate triplets.
The order-27 subgroup of SU(3) generated by diag(ω,1,ω⁻¹), diag(1,ω,ω⁻¹) and the cyclic coordinate permutation, with ω³ = 1 and ω ≠ 1.
Definition sourcesReference properties
Representation scope
The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities. [sources]
Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue. [sources]
| Irrep label | Complex dimension | Reality type |
|---|---|---|
| ρ1 | 1 | Real |
| ρ2 | 1 | Complex |
| ρ3 | 1 | Complex |
| ρ4 | 1 | Complex |
| ρ5 | 1 | Complex |
| ρ6 | 1 | Complex |
| ρ7 | 1 | Complex |
| ρ8 | 1 | Complex |
| ρ9 | 1 | Complex |
| ρ10 | 3 | Complex |
| ρ11 | 3 | Complex |
Structure and conventions
The center and commutator subgroup are both C₃; the abelianization is C₃ × C₃. [sources]
The SU(3) matrix realization is used in the cited study of invariant rings and orbifold gauge theories. [sources]
Names and aliases
Delta(27) · He3 · Heisenberg group of order 27 · 3_+^(1+2) · (C3 × C3) ⋊ C3
Related objects and maps
- Specified SU(3) embedding
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 order-27 subgroup of SU(3) generated by diag(ω,1,ω⁻¹), diag(1,ω,ω⁻¹) and the cyclic coordinate permutation, with ω³ = 1 and ω ≠ 1.
- Counting BPS Operators in Gauge Theories: Quivers, Syzygies and PlethysticsEquation (3.12)Open source
- GroupNames: He₃Heading and AliasesOpen source
group:delta-27#definitionOrdinary complex irreducibles
1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3
Assumptions: Characteristic zero · Finite-dimensional irreducibles over C · Dimensions listed with multiplicity
- GroupNames: He₃Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:delta-27#irrepsGAP SmallGroups ID
SmallGroup(27,3)
- GroupNames: He₃Aliases; GAP/Magma code by IDOpen source
group:delta-27#smallgroups-idRepresentation scope
The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities.
- GroupNames: He₃Character tableOpen source
group:delta-27#representations-1Representation scope
Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue.
- GroupNames: He₃Character table: identity column and real/complex/symplectic annotationsOpen source
- GAP · Characters and symmetrizationsIrr and section 71.12-5 IndicatorOpen source
group:delta-27#representations-2Structure and conventions
The center and commutator subgroup are both C₃; the abelianization is C₃ × C₃.
- GroupNames: He₃Centre; CommutatorOpen source
group:delta-27#structure-1Structure and conventions
The SU(3) matrix realization is used in the cited study of invariant rings and orbifold gauge theories.
- Counting BPS Operators in Gauge Theories: Quivers, Syzygies and PlethysticsSection 3, equation (3.12) and following discussionOpen source
group:delta-27#structure-2Specified SU(3) embedding
group:su-3
- Counting BPS Operators in Gauge Theories: Quivers, Syzygies and PlethysticsEquation (3.12)Open source
group:delta-27#relation-1