The Peter–Weyl theorem is a fundamental result in the representation theory of compact topological groups. It describes how the regular representation of a compact group can be decomposed into irreducible representations. Here's a brief overview of the main points of the theorem: 1. **Compact Groups**: The theorem applies specifically to compact groups, which are groups that are also compact topological spaces. Examples include \(SU(n)\), \(SO(n)\), and \(U(n)\).
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For a compact group, the matrix coefficients of its finite-dimensional irreducible unitary representations span a dense subspace of and form a complete orthogonal decomposition of with Haar measure. If is a closed subgroup and is a finite-dimensional unitary -representation, the space of L2 sections of the associated homogeneous bundle has multiplicity for each irreducible -representation . This is the compact-group form of Frobenius reciprocity and determines homogeneous-bundle multiplicities from isotropy weights.