Example

Take P equal to $A_s$. For $\lambda \in \mathscr{S}$, denote the opposite field of the commutant of the simple A-module $S_{\lambda}$ by $D_{\lambda}$ and the dimension of $S_{\lambda}$ viewed as a right vector space over the field $D_{\lambda}$ by $m(\lambda )$. We know that $m(\lambda )$ is equal to the multiplicity $[A_s/\mathfrak{r}A_s:\lambda ]$ (VIII, p. 143, Proposition 11). Consequently, the A-module $A_s$ is isomorphic to $\oplus_{\lambda\in\mathscr{S}}P^{m(\lambda)}_{\lambda}$.