Finite groups · Representation
S₄ irreducible representation sgn
A 1-dimensional ordinary complex irrep of S₄, of real type, with an exact character and executable tensor operations.
Executable coverageOrdinary complexCharacterIrreducible
Definition
The ordinary complex irreducible representation labeled sgn of S₄, specified by its character on the named conjugacy classes. Sign of a permutation
Definition sourcesRepresentation identity
Reference properties
Character and identity
This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation. [sources]
Tensor operations and physical use
Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients. [sources]
Conventions and realization
The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1. [sources]
Names and aliases
S₄ irrep sgn
Related objects and maps
- Representation domain
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 ordinary complex irreducible representation labeled sgn of S₄, specified by its character on the named conjugacy classes. Sign of a permutation
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#definitionComplex dimension
1
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#dimensionCoefficient field
ℂ; characteristic zero
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#fieldFrobenius–Schur type
Real
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#realityFrobenius–Schur indicator
1
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#indicatorComplex conjugate irrep
sgn
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#conjugateCharacter and identity
This is an abstract ordinary complex representation. Its character, rather than its dimension alone, fixes its identity within the specified group and class convention. Columns use explicit cycle representatives, ordered 1,(12),(12)(34),(123),(1234). Here 3 is the standard permutation-minus-trivial representation; 3′ = sgn ⊗ 3 is the orientation-preserving cubic-rotation representation.
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#representations-1Tensor operations and physical use
Use the linked character workbench to count internal singlets and decompose tensor, symmetric, or exterior powers. Field statistics apply to the full label space, including Lorentz and species labels when present. A character decomposition does not specify normalized invariant tensors or Clebsch–Gordan coefficients.
- Etingof et al., Introduction to Representation TheorySections 4.4–4.5, 4.8–4.9, and 5.1: characters, orthogonality, small-group tables, tensor products, and Frobenius–Schur indicatorsOpen source
- GAP · Characters and symmetrizationsSections 72.7–72.8 and 72.11: scalar products, decompositions, symmetric and exterior powersOpen source
representation:s-4-sgn#physics-1Conventions and realization
The coefficient field is complex; real describes the Frobenius–Schur type of this complex irrep. Labels refer to the table published here, not an upstream row number. No matrix basis is selected. Convention profile: finite-ordinary-complex-v1.
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#conventions-1Representation domain
group:s-4
- GroupNames: S₄Ordinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- Etingof et al., Introduction to Representation TheoryOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Characters and symmetrizationsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP · Character tables and Frobenius–Schur indicatorsOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
- GAP Reference Manual, Chapter 73: Maps Concerning Character TablesOrdinary character table of S₄; the irrep sgn is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source
representation:s-4-sgn#relation-1