Peter–Weyl Theorem — Mathematics Knowledge System

Overview

The Peter–Weyl theorem is the fundamental structure theorem for the Hilbert space L²(G) of square-integrable functions on a compact topological group G with Haar measure. It states that L²(G) decomposes as a completed orthogonal direct sum of matrix coefficient spaces, one for each irreducible unitary representation. As a consequence, the matrix coefficients of irreducible representations form an orthonormal basis of L²(G), and every irreducible unitary representation is finite-dimensional. The theorem generalises the theory of Fourier series (G = U(1)) to all compact groups.

Intuition

For G = U(1) = {e^{iθ}}, L²(G) = L²[0,2π] and the Fourier series decomposition Σ aₙe^{inθ} is exactly Peter–Weyl: each 1-dimensional rep e^{inθ} contributes one matrix coefficient. For a non-abelian compact group, you get blocks of size dim(ρ)² for each irreducible ρ, encoding all (dim ρ)² matrix coefficients.

Formal Definition

Definition

Let G be a compact topological group with normalised Haar measure μ (μ(G)=1). Let {(ρ_α, V_α)} be a complete set of pairwise inequivalent irreducible unitary representations. The matrix coefficient functions are

c^\alpha_{ij}(g) = \langle \rho_\alpha(g) e_j,\; e_i \rangle \quad (\text{for orthonormal basis } \{e_i\} \text{ of } V_\alpha)

Matrix coefficient functions

L^2(G) = \widehat{\bigoplus}_{\alpha} \mathrm{End}(V_\alpha) \cong \widehat{\bigoplus}_{\alpha} V_\alpha \otimes V_\alpha^*

Peter–Weyl decomposition of L²(G)

\langle c^\alpha_{ij},\; c^\beta_{kl} \rangle_{L^2(G)} = \frac{\delta_{\alpha\beta}\,\delta_{ik}\,\delta_{jl}}{\dim V_\alpha}

Orthogonality of matrix coefficients

f = \sum_\alpha \dim(V_\alpha) \sum_{i,j} \langle f, c^\alpha_{ij}\rangle\, c^\alpha_{ij} \quad \text{in } L^2(G)

Expansion of f ∈ L²(G) in matrix coefficient basis

Notation

Notation	Meaning
$L^2(G)$	Hilbert space of square-integrable functions on G with Haar measure
$c^\alpha_{ij}$	Matrix coefficient of ρ_α: c^α_{ij}(g) = ⟨ρ_α(g)eⱼ, eᵢ⟩
$\hat{G}$	Set of equivalence classes of irreducible unitary representations of G
$\mu$	Normalised Haar measure on G

Properties

Character orthogonality

⟨χ_α, χ_β⟩_{L²(G)} = δ_{αβ} — characters of distinct irreducible reps are orthonormal.

Plancherel formula

‖f‖²_{L²(G)} = Σ_α dim(V_α) ‖f̂(α)‖²_{HS}, where ‖·‖_HS is the Hilbert-Schmidt norm on End(V_α).

Theorems

Theorem 1: Peter–Weyl Theorem

For a compact group G, the matrix coefficient functions {√(dim V_α) · c^α_{ij}} form a complete orthonormal system in L²(G). Thus L²(G) ≅ ⊕̂_α V_α ⊗ V_α* (Hilbert-space direct sum).

Theorem 2: Finite-Dimensionality of Irreducibles

Every irreducible unitary representation of a compact group is finite-dimensional.

Theorem 3: Fourier Inversion

For f ∈ C(G) (continuous): f(g) = Σ_α dim(V_α) tr(ρ_α(g) f̂(α)), where f̂(α) = ∫_G f(g) ρ_α(g)* dg ∈ End(V_α).

Worked Examples

1
G = U(1) = {e^{iθ}} with Haar measure dθ/2π. Irreducible unitary representations: ρₙ(e^{iθ}) = e^{inθ} for n ∈ ℤ, each 1-dimensional.
$\hat{U}(1) = \{\rho_n : n \in \mathbb{Z}\}, \quad \rho_n(e^{i\theta}) = e^{in\theta}$
2
Matrix coefficients (1×1 matrices): c^n_{11}(e^{iθ}) = e^{inθ}. Peter–Weyl says these form an orthonormal basis of L²(U(1)).
$\langle c^m_{11}, c^n_{11}\rangle = \int_0^{2\pi} e^{im\theta} e^{-in\theta} \frac{d\theta}{2\pi} = \delta_{mn}$
3
This is exactly the classical Fourier orthonormality of {e^{inθ}}_{n∈ℤ}, and their completeness in L²[0,2π] is the Fourier series theorem.
$L^2(U(1)) = \widehat{\bigoplus}_{n \in \mathbb{Z}} \mathbb{C} \cdot e^{in\theta}$

✓ Answer

For U(1), Peter–Weyl reduces to the classical Fourier series. The irreducibles are {e^{inθ}}_{n∈ℤ} and they form an ONB of L²(U(1)).

Practice Problems

Mediumfree response

State the orthogonality relations for matrix coefficients and derive the character orthogonality ⟨χ_α, χ_β⟩_G = δ_{αβ} from them.

Hardfree response

Why does the Peter–Weyl theorem imply that irreducible unitary representations of compact groups are finite-dimensional?

Quiz

Summary

Peter–Weyl: L²(G) ≅ ⊕̂_α V_α ⊗ V_α* as a Hilbert-space direct sum over all irreducibles.
Matrix coefficients of irreducible representations form an orthogonal basis of L²(G).
Orthogonality: ⟨c^α_{ij}, c^β_{kl}⟩ = δ_{αβ}δ_{ik}δ_{jl}/dim V_α.
Every irreducible unitary representation of a compact group is finite-dimensional.
For G = U(1) this is classical Fourier series; for SU(2) it is expansion in spin harmonics.

References

BookBröcker, T. & tom Dieck, T. — Representations of Compact Lie Groups (1985), Chapter 4
BookFolland, G.B. — A Course in Abstract Harmonic Analysis (1995), §5.1