Orthogonality Relations

Core Idea

The irreducible characters of a finite group G are orthonormal with respect to the inner product ⟨χ, ψ⟩ = (1/|G|) Σ_{g∈G} χ(g) conjugate(ψ(g)). This means ⟨χᵢ, χⱼ⟩ = δᵢⱼ — different irreducible characters are orthogonal, and each has norm 1. These relations, derived from Schur's lemma, are the primary computational tool for decomposing representations and constructing character tables.

Explainer

The orthogonality relations are the quantitative backbone of character theory. Define an inner product on the space of class functions (functions G → ℂ that are constant on conjugacy classes) by ⟨f₁, f₂⟩ = (1/|G|) Σ_{g∈G} f₁(g) conjugate(f₂(g)). The first orthogonality relations (row orthogonality) state that the irreducible characters χ₁, …, χₖ form an orthonormal set: ⟨χᵢ, χⱼ⟩ = δᵢⱼ.

The proof distills from Schur's lemma. Given irreducible representations ρ and σ, consider the "averaged" operator T̃ = (1/|G|) Σ_{g∈G} σ(g)⁻¹ T ρ(g) for an arbitrary linear map T. Schur's lemma forces T̃ to be zero when ρ ≇ σ, and a scalar when ρ ≅ σ. Taking traces with judicious choices of T yields the orthogonality relations. The proof is constructive — it builds the intertwining operators whose properties Schur's lemma constrains.

The practical payoff is enormous. To decompose a representation V into irreducibles, write χ_V = n₁χ₁ + ··· + nₖχₖ. Taking inner products: nᵢ = ⟨χ_V, χᵢ⟩. Each inner product is a finite sum over the group (or equivalently, a weighted sum over conjugacy classes). To test irreducibility: compute ⟨χ, χ⟩ = Σ nᵢ²; the result is 1 if and only if the representation is irreducible. These are concrete, computable checks.

There are also column orthogonality relations, obtained by summing over irreducible representations rather than group elements: Σᵢ χᵢ(C_r) conjugate(χᵢ(C_s)) = |G|/|C_r| · δᵣₛ, where C_r, C_s are conjugacy classes. Together, the row and column relations impose so many constraints on a character table that it can often be determined with minimal additional input. The number of irreducible characters equals the number of conjugacy classes, so the character table is always square — a fact that underscores the deep duality between group elements and representations.