Group representation theory MOC

Decomposition of a representation

By Maschke's theorem, a representation of a compact group on a finite vector space may be decomposed into the direct sum of irreps.

• An irrep may be carried by one or more mutually orthogonal (irreducible) invariant subspaces (where distinguishes multiplicities)
• These subspaces are given Irreducible orthonormal basis
• We then denote concrete reälizations of each acting on this subspace by , however the can be dropped when referring to matrix entries since these can be selected to be the same in all repeat subspaces.

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