Characters and Character Theory — Mathematics Knowledge System

Overview

The character of a representation ρ: G → GL(V) is the function χ_ρ: G → k defined by χ_ρ(g) = tr(ρ(g)). Characters are class functions (constant on conjugacy classes) and contain enough information to determine the isomorphism class of the representation over algebraically closed fields of characteristic 0. The orthogonality relations for characters are among the most powerful tools in representation theory of finite groups.

Intuition

The character χ(g) = tr(ρ(g)) is a single number attached to each group element. Because the trace is invariant under similarity (tr(ABA⁻¹) = tr(B)), the character depends only on the conjugacy class of g. Two representations with the same character are isomorphic — so the character is a complete invariant of the (complex) representation.

Formal Definition

Definition

Let ρ: G → GL(V) be a representation over k. The character of ρ is the function χ_ρ: G → k defined by the trace. Inner products of characters are defined using averaging over the group.

\chi_\rho(g) = \mathrm{tr}(\rho(g)) \in k

Character function

\chi_\rho(hgh^{-1}) = \chi_\rho(g) \quad \forall g,h \in G

Character is a class function

\langle \chi, \psi \rangle = \frac{1}{|G|}\sum_{g \in G} \chi(g)\,\overline{\psi(g)}

Inner product on class functions (over ℂ)

\langle \chi_V, \chi_W \rangle = \dim_\mathbb{C} \mathrm{Hom}_G(V, W)

Inner product counts intertwining maps

Notation

Notation	Meaning
$\chi_\rho,\; \chi_V$	Character of representation ρ or G-module V
$\langle \chi, \psi \rangle$	Inner product of class functions
$\mathrm{Cl}(G)$	Set of conjugacy classes of G

Properties

Character of direct sum

\chi_{V \oplus W} = \chi_V + \chi_W

Character of tensor product

\chi_{V \otimes W} = \chi_V \cdot \chi_W

Character of dual

\chi_{V^*}(g) = \overline{\chi_V(g)}

Dimension from character

\dim V = \chi_V(e)

Theorems

Theorem 1: First orthogonality relation

\langle \chi_i, \chi_j \rangle = \delta_{ij} \quad \text{for distinct irreducible characters } \chi_i, \chi_j

Theorem 2: Second orthogonality relation

\sum_{\rho \text{ irred.}} \chi_\rho(g)\,\overline{\chi_\rho(h)} = \begin{cases}|C_G(g)| & g \sim h \\ 0 & \text{otherwise}\end{cases}

Theorem 3: Character determines representation

Two complex representations of a finite group are isomorphic if and only if they have the same character.

Theorem 4: Number of irreducibles

The number of irreducible complex representations of G equals the number of conjugacy classes of G.

Worked Examples

1
S₃ has 3 conjugacy classes: {e}, {(12),(13),(23)}, {(123),(132)}.
$\mathrm{Cl}(S_3) = \{\{e\},\{\text{transpositions}\},\{\text{3-cycles}\}\}$
2
The standard representation V is ℝ³/⟨(1,1,1)⟩, dimension 2. Permutation matrices restricted to V.
3
χ(e) = 2 (trace of I₂); χ((12)) = tr of the transposition matrix on V = 0; χ((123)) = tr of 3-cycle on V = -1.
$\chi(e) = 2,\; \chi((12)) = 0,\; \chi((123)) = -1$

✓ Answer

Character values: 2 on {e}, 0 on transpositions, -1 on 3-cycles.

Practice Problems

Mediumfree response

Decompose the permutation representation of S₃ on ℂ³ into irreducibles using characters.

Hardproof writing

Prove that Σᵢ (dim Vᵢ)² = |G|, where the sum is over all irreducible complex representations.

Common Mistakes

Common Mistake

Thinking the character determines the representation over any field

Characters determine representations only over algebraically closed fields of characteristic 0 (such as ℂ). Over ℝ or finite fields, non-isomorphic representations can have the same character.

Common Mistake

Confusing the character with the determinant

The character is the trace; the determinant is also a class function but contains less information. The trace records all eigenvalues with multiplicity; the determinant is only their product.

Quiz

The character χ_V(g) of a representation V is defined as:

det(ρ(g))tr(ρ(g))dim VThe minimal polynomial of ρ(g)

The number of irreducible complex representations of a finite group G equals:

The order |G|The number of conjugacy classesThe number of subgroupsdim Z(kG)

Historical Background

Frobenius introduced characters in 1896 while factoring the group determinant det(x_{g h⁻¹}) for a finite group G. He proved the fundamental orthogonality relations without initially having the concept of a representation itself (that came slightly later via Schur and Burnside). The connection between characters and the trace of matrices was made explicit by Burnside. The theory reached its modern form with Schur's orthogonality relations and the complete character theory of the symmetric group.

1896
Frobenius defines characters of finite groups via the group determinant
Georg Frobenius
1897
Frobenius proves orthogonality of characters
Georg Frobenius
1905
Schur proves Schur's lemma, completing the framework
Issai Schur
1900s
Burnside applies character theory to prove burnside's p^a q^b theorem
William Burnside

Summary

The character χ_ρ(g) = tr(ρ(g)) is a class function on G.
Characters are additive under direct sum and multiplicative under tensor product.
First orthogonality: irreducible characters form an orthonormal set; second orthogonality involves summing over irreducibles.
Over ℂ, two representations are isomorphic iff they have the same character.
The number of irreducible complex representations equals the number of conjugacy classes.

References

BookSerre, J.-P. — Linear Representations of Finite Groups (1977), Chapters 2–3
BookIsaacs, I.M. — Character Theory of Finite Groups (1976), Chapters 1–2
WebsiteWikipedia — Character theory
WebsiteMathWorld — Group Character