Orthogonality and the Regular Representation

The inner product on class functions

Put a Hermitian inner product on functions G → C by ⟨α, β⟩ = (1/|G|) Σ_{g∈G} α(g) · conjugate(β(g)). The first orthogonality relation is the cornerstone: the irreducible characters are orthonormal, ⟨χᵢ, χⱼ⟩ = δᵢⱼ (1 if i = j, else 0). Since characters are class functions and there is one irreducible per conjugacy class, the irreducible characters form an orthonormal basis of the space of class functions.

Where does it come from? Take irreducibles V, W. Average any linear map f: V → W to get f₀ = (1/|G|) Σ_g ρ_W(g) f ρ_V(g)⁻¹, which is equivariant. By Schur, f₀ = 0 when V ≇ W and f₀ = (tr f / dim V)·id when V = W. Taking matrix entries of these two cases and summing gives exactly ⟨χ_V, χ_W⟩ = δ. Orthogonality is Schur's lemma with the bookkeeping done.

Decomposing by inner products

Now any representation V decomposes as ⊕ mᵢVᵢ, and the multiplicity is just an inner product: mᵢ = ⟨χ_V, χᵢ⟩. To analyze a representation you no longer chase invariant subspaces — you compute one number per irreducible. This is the practical payoff of the whole theory.

1. Compute χ_V on each conjugacy class — often just counting fixed points if V is a permutation representation.
2. For each irreducible χᵢ in the character table, compute mᵢ = ⟨χ_V, χᵢ⟩ by the weighted sum over classes.
3. Read off V ≅ ⊕ mᵢVᵢ; sanity-check Σ mᵢ·(dim Vᵢ) = dim V and Σ mᵢ² = ⟨χ_V, χ_V⟩.

The regular representation

Let G act on the group algebra C[G] itself by left multiplication — the regular representation. Its character is dramatic: χ_reg(e) = |G| and χ_reg(g) = 0 for g ≠ e, because left-multiplication by g ≠ e is a fixed-point-free permutation of the basis G. Feeding this into mᵢ = ⟨χ_reg, χᵢ⟩ gives mᵢ = (1/|G|)·|G|·χᵢ(e) = dim Vᵢ. So the regular representation contains every irreducible, each with multiplicity equal to its own degree.

Counting dimensions in the regular representation of S_3.

C[S_3] has dimension |S_3| = 6.
Regular rep decomposes as  reg = (dim V_i) copies of each V_i:
  reg  =  1*trivial  (+)  1*sign  (+)  2*standard
dim check:  1*1 + 1*1 + 2*2  =  1 + 1 + 4  =  6.  OK

Reading the e-column gives the famous identity:
  SUM over irreducibles of (degree)^2  =  |G|.
For S_3:  1^2 + 1^2 + 2^2 = 6.

This identity, plus #irreducibles = #conjugacy classes, often pins down
the ENTIRE list of degrees by hand. E.g. a group of order 8 with 5
classes must have degrees d satisfying  d_1^2+...+d_5^2 = 8  with five
positive integers => 1,1,1,1,2  (four linear chars and one 2-dim).