Decomposition of a representation

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

An irrep $Γ 𝜇$ may be carried by one or more mutually orthogonal (irreducible) invariant subspaces $𝑉 𝜇 𝛼$ (where $𝛼$ distinguishes multiplicities)
These subspaces are given Irreducible orthonormal basis $𝑒 𝜇 𝛼 𝑗$
We then denote concrete reälizations of each $Γ 𝜇$ acting on this subspace by $Γ 𝜇 𝛼$ , however the $𝛼$ 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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