Isaacs, $\textit{Character Theory of Finite Groups}$, Corollary(1.17)
Let $A$ be a semisimple algebra over an algebraically closed field $F$ and let $M$ be an irreducible $A$-module. Then
- $A_M=End(M)$;
- $dim(A_M)=dim(M(A))=dim(M)^2$;
- $n_M(A^\circ)=dim(M)$.
- $dim(A)=\sum_{M\in\mathcal{M}(A)}dim(M)^2$,
- $dim(Z(A))=|\mathcal{M}(A)|$.
Pf:
- $D=E_A(M)=F\cdot1$, $A_M=E_D(M)=End(M)$, so we prove 1,2,3
- $A=\sum\cdot_{M\in\mathcal{M}(A)}M(A)$. 4 is immediate from 2.
- $Z(A_M)=A_M\cap E_A(M)=A_M\cap F\cdot1=F\cdot1$. Thus $dim(Z(M(A)))=dim(Z(A_M))=1$.
- $Z(A)=\sum_{M\in\mathcal(M)}Z(M(A))$.