Math 2.0

Group theory and generalizations Definition

Character of a representation

Builds on Group representationTrace of a matrixConjugacy classDimension

Definition

Let GG be a group and let FF be a field. Let ρ ⁣:G→GL(V)\rho \colon G \to GL(V) be a representation of GG on a vector space VV over FF of finite dimension. The character of ρ\rho is the function χρ ⁣:G→F\chi_\rho \colon G \to F given by χρ(g)=tr⁡ρ(g),\chi_\rho(g) = \tr \rho(g), where tr⁡\tr is the trace of a linear map. The degree of χρ\chi_\rho is dim⁡V\dim V. When FF has characteristic 00, the degree equals χρ(1)\chi_\rho(1); in characteristic pp the value χρ(1)\chi_\rho(1) is only dim⁡V\dim V modulo pp. A character is irreducible if ρ\rho is irreducible.

Remarks

Example: let S3S_3 act on ℂ3\C^3 by permuting coordinates. The matrix of a permutation σ\sigma has a 11 on the diagonal exactly at the points fixed by σ\sigma, so χ(σ)\chi(\sigma) is the number of fixed points of σ\sigma. This gives χ(1)=3\chi(1) = 3, χ(transposition)=1\chi(\text{transposition}) = 1 and χ(3-cycle)=0\chi(\text{3-cycle}) = 0. Non-example: let ℤ\Z act by shifting on the space of finitely supported sequences. This space has infinite dimension, so the trace is not defined, and the representation has no character. Every character satisfies χ(hgh−1)=χ(g)\chi(hgh^{-1}) = \chi(g), so it is constant on each conjugacy class; such functions are called class functions. Isomorphic representations have the same character. For a finite group over ℂ\C, the converse holds as well.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Conjugacy class, concept, no evidence levelConjugacy classconceptDimension, concept, no evidence levelDimensionconceptGroup representation, concept, no evidence levelGroup representationconceptTrace of a matrix, concept, no evidence levelTrace of a matrixconceptCharacter of a representation, definition, no evidence levelCharacter of a repres…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Conjugacy class concept
  2. Dimension concept
  3. Group representation concept
  4. Trace of a matrix concept
  5. Character of a representation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Character_theory
About this page
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MSC20C15
DefinesCharacter of a representation
UsesGroup representation, Trace of a matrix, Conjugacy class, Dimension
Changed2 November 2026

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