REFERENCE SHEET

Characters & decomposition

Traces summarize finite-group representations without fixing a basis.

Before this Representations & invariant subspaces, Eigenvalues & diagonalization

⟨ χ, ψ ⟩ = \frac{1}{| G |} \sum_{g \in G} χ (g) \bar{ψ (g)}

Definitions

Character

χV(g)=tr ρV(g). This sheet assumes finite groups and finite-dimensional complex representations.

Class function

A function constant on conjugacy classes. Characters are class functions because trace is unchanged by conjugation.

Multiplicity

The number of copies of an irreducible representation in a direct-sum decomposition.

Key theorems

Character orthogonality

Irreducible complex characters of a finite group form an orthonormal basis of its class functions under the displayed inner product.

Multiplicity formula

If χ is the character of V and χᵢ is irreducible, its multiplicity in V is ⟨χ,χᵢ⟩.

Dimension sum

The irreducible complex representation dimensions dᵢ satisfy Σᵢdᵢ²=|G|.

Notable applications

Decomposition without eigenvectors

Character inner products recover multiplicities without explicitly finding invariant bases.

Permutation actions

The trace of a permutation matrix equals its number of fixed basis elements, linking characters to orbit counting.

Work it out

Try before revealing

01 / Compute

For C₂={e,s}, a representation has character (3,1). Decompose it using irreducible characters (1,1) and (1,−1).

Hint

Compute the two inner products.

Solution

The trivial multiplicity is (3+1)/2=2; the sign multiplicity is (3−1)/2=1.

02 / Compute

Find the character of the natural S₃ permutation action on ℂ³ at e, (12), and (123).

Hint

Count fixed coordinates.