Introduction to Representation Theory
《Introduction to Representation Theory》 · 第 120 页 · Chapter 5，Section 5.18「Schur-Weyl duality for $\mathfrak{gl}(V)$」，Theorem 5.18.1
Theorem 5.18.1. Let $A, B$ be two subalgebras of the algebra $\operatorname{End} E$ of endomorphisms of a finite dimensional vector space $E$, such that $A$ is semisimple and $B = \operatorname{End}_A E$. Then: (i) $A = \operatorname{End}_B E$ (i.e., the centralizer of the centralizer of $A$ is $A$); (ii) $B$ is semisimple; (iii) As a representation of $Aotimes B$, $E$ decomposes as $E = \bigoplus_{i\in I} V_i otimes W_i$, where $V_i$ are all the irreducible representations of $A$ and $W_i$ are all the irreducible representations of $B$.