Character Theory

Core Idea

The character of a representation ρ: G → GL(V) is the function χ_ρ: G → F defined by χ_ρ(g) = tr(ρ(g)). Characters are class functions (constant on conjugacy classes), are independent of the choice of basis, and — remarkably — determine the representation up to equivalence over ℂ. Character theory reduces the study of representations to the study of these numerical functions, making computation tractable.

Explainer

Character theory is the computational engine of representation theory for finite groups. The character of a representation ρ: G → GL(V) is the function χ_ρ: G → ℂ defined by χ_ρ(g) = tr(ρ(g)) — the trace of the matrix (in any basis) representing g. The trace is basis-independent (since tr(PAP⁻¹) = tr(A)) and satisfies tr(AB) = tr(BA), which makes characters constant on conjugacy classes: χ(hgh⁻¹) = tr(ρ(h)ρ(g)ρ(h)⁻¹) = tr(ρ(g)) = χ(g).

The first key property is additivity: if V = W₁ ⊕ W₂, then χ_V = χ_{W₁} + χ_{W₂}. This follows from the trace of a block-diagonal matrix being the sum of the block traces. The second key property, far deeper, is faithfulness: over ℂ, two representations with the same character are equivalent. This means the character — a simple numerical function — captures all the information in the representation up to isomorphism.

To decompose a representation V into irreducibles V₁, …, Vₖ with multiplicities n₁, …, nₖ, we have χ_V = n₁χ₁ + ··· + nₖχₖ. The orthogonality relations (the next topic) provide an inner product on class functions under which the irreducible characters form an orthonormal basis. The multiplicity nᵢ is then simply the inner product ⟨χ_V, χᵢ⟩ — a finite computation involving a sum over the group. This transforms the algebraic problem of decomposing a representation into a numerical calculation.

The number of distinct irreducible characters equals the number of conjugacy classes of G. For S₃, there are 3 conjugacy classes and 3 irreducible characters. For the symmetric group Sₙ, the number of conjugacy classes equals the number of partitions of n, connecting representation theory to combinatorics. The character table — a square matrix whose rows are irreducible characters and whose columns are conjugacy classes — encodes the entire representation theory of a finite group in a compact, computable form.