Peter-Weyl theorem

The Peter-Weyl theorem is the bridge that carries finite group character theory over to compact groups — continuous symmetry groups like the circle or the rotation group SO(3). It says that even though such a group is infinite, you can still build all its representation theory from finitely-many-dimensional irreducible pieces, and the matrix entries of these pieces are rich enough to approximate any reasonable function on the group.

Let G be a compact (Hausdorff topological) group with its Haar measure, the canonical translation-invariant probability measure. The Peter-Weyl theorem has several intertwined parts. First, every finite-dimensional representation is completely reducible and every irreducible unitary representation is finite-dimensional. Second, the matrix coefficients of the irreducible representations — functions of the form g -> (an entry of rho(g)) — span a dense subspace of the continuous functions C(G) in the uniform norm. Third, and most precisely, L^2(G) decomposes as the Hilbert space direct sum over the irreducibles rho of (dim rho) copies of rho, with an orthonormal basis given by the scaled matrix coefficients.

This generalizes the finite case exactly: there the group algebra C[G] plays the role of L^2(G), Haar measure is the normalized counting measure, and the decomposition is the Artin-Wedderburn decomposition of the regular representation. For the circle group U(1) the irreducibles are the characters z -> z^n for integers n, and Peter-Weyl specializes to the completeness of the Fourier basis — classical Fourier series are the Peter-Weyl theorem for the simplest compact group.