Introduction to Mathematical Logic, Sixth Edition
《Introduction to Mathematical Logic, Sixth Edition》 · 印刷第 426 页 Section 10.1.1「Centralizer Algebra」，张量幂表示的对象约定、作用公式与中心化子代数定义
Let $V$ be a finite-dimensional complex vector space and let $G$ be a reductive algebraic subgroup of $\operatorname{GL}(V)$. Denote by $\rho_k$ the natural representation of $G$ on $\bigotimes^k V$: $\rho_k(g)(v_1\otimes\cdots\otimes v_k)=gv_1\otimes\cdots\otimes gv_k$ for $v_i\in V$. To determine the decomposition of $\bigotimes^k V$ into $G$-isotypic components, we can use the approach of the previous chapter for $\operatorname{GL}(V)$. Namely, we form the centralizer algebra $\mathcal B=\{B\in\operatorname{End}(\bigotimes^k V):B\rho_k(g)=\rho_k(g)B\text{ for all }g\in G\}$.