Finite groups · Representation

$Q_8:\quad \mathbf{1}$ Q₈ irreducible representation 1

A 1-dimensional ordinary complex irrep of Q₈, of real type, with an exact character and executable tensor operations.

Executable coverageOrdinary complexCharacterIrreducible

Definition

The ordinary complex irreducible representation labeled 1 of Q₈, specified by its character on the named conjugacy classes. Trivial representation

Definition sources

Complex dimension: $1$
Coefficient field: ℂ; characteristic zero

Representation identity

Domain

\mathrm{Q}_{8}

Categoryordinary-complex

Coefficient field

\mathbb C

Conventionfinite-ordinary-complex-v1

Character on the named conjugacy classes
$1$	$-1$	$i$	$j$	$k$
$1$	$1$	$1$	$1$	$1$

This record specifies an abstract representation. A matrix basis and normalized invariant tensors are separate choices.

Reference properties

Property	Value	Evidence
Complex dimension	$1$	Sources
Coefficient field	ℂ; characteristic zero	Sources
Frobenius–Schur type	Real	Sources
Frobenius–Schur indicator	$1$	Sources
Complex conjugate irrep	$1$	Sources

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 are 1, -1, {±i}, {±j}, {±k}. Linear labels a,b,c refer respectively to signs (+,-),(-,+),(-,-) on (i,j). [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

Q₈ irrep 1

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathrm{Q}_{8}$
Quaternion group
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 1 of Q₈, specified by its character on the named conjugacy classes. Trivial representation

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#definition

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Complex dimension

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#dimension

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Coefficient field

ℂ; characteristic zero

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#field

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Frobenius–Schur type

Real

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#reality

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Frobenius–Schur indicator

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#indicator

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Complex conjugate irrep

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#conjugate

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Character and identity

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#representations-1

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Tensor operations and physical use

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:q-8-1#physics-1

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Conventions and realization

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#conventions-1

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Representation domain

group:q-8

GroupNames: Q₈Ordinary character table of Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 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 Q₈; the irrep 1 is identified by the complete character vector in this record. Local class representatives and labels are supplied with the table.Open source

representation:q-8-1#relation-1

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Q8:1Q_8:\quad \mathbf{1}Q8​:1Q₈ irreducible representation 1

Representation identity

Reference properties

Character and identity

Tensor operations and physical use

Conventions and realization

Names and aliases

Related objects and maps

Claims and sources

$Q_8:\quad \mathbf{1}$ Q₈ irreducible representation 1