每日定理10

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

1. $A_M=End(M)$;
2. $dim(A_M)=dim(M(A))=dim(M)^2$;
3. $n_M(A^\circ)=dim(M)$.
4. $dim(A)=\sum_{M\in\mathcal{M}(A)}dim(M)^2$,
5. $dim(Z(A))=|\mathcal{M}(A)|$.

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 $dim(Z(M(A)))=dim(Z(A_M))=1$.
• $Z(A)=\sum_{M\in\mathcal(M)}Z(M(A))$.