[[Group representation theory MOC]]
# Decomposition of a representation

By [[Maschke's theorem]], a representation $\Gamma$ of a compact group $G$ on a finite vector space $V$ may be decomposed into the [[Direct sum of representations|direct sum]] of [[Irrep|irreps]].

- An irrep $\Gamma^\mu$ may be carried by one or more mutually orthogonal (irreducible) invariant subspaces $V^{\mu\alpha}$ (where $\alpha$ distinguishes multiplicities)
- These subspaces are given [[Irreducible orthonormal basis]] $e^{\mu\alpha}_{j}$
- We then denote concrete reälizations of each $\Gamma^\mu$ acting on this subspace by $\Gamma^{\mu\alpha}$, however the $\alpha$ 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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