Semisimple representation

A representation is semisimple when it is a direct sum of irreducible representations. Equivalently, every invariant subspace has an invariant complement.

For a finite-dimensional complex semisimple representation $V={⨁}_{\lambda }({\mathbb{C}}^{m_{\lambda }}\otimes V_{\lambda })$ with pairwise inequivalent irreducible representations $V_{\lambda }$, its commuting endomorphisms are exactly ${⨁}_{\lambda }(A_{\lambda }\otimes I_{V_{\lambda }})$. To prove it, write each block as an intertwiner and apply Schur's lemma: blocks between inequivalent constituents vanish, and blocks between fixed copies of the same constituent are scalar. The dimension is ${\sum }_{\lambda }{m}_{\lambda }^2$.