Proposition 11

Let M be a semisimple A-module that is the direct sum of a family $(M_i)_{i\in I}$ of simple submodules. For every $\lambda \in \mathscr{S}$, we denote by $I(\lambda )$ the set of indices $i\in I$ such that $M_i$ is of class $\lambda$. The cardinal of $I(\lambda )$ is equal to the dimension of the left $D_{\lambda}$-vector space Hom$_A(S_{\lambda},M)$.

The isotypical component of A of type $\lambda$ is isomorphic to $S_{\lambda}^{(I(\lambda))}$ (VIII, p. 65, Proposition 4, b)). The $D_{\lambda}$-vector space Hom$_A(S_{\lambda},M)$ may be identified with Hom$_A(S_{\lambda},M_{\lambda})$, so is isomorphic to $D_{\lambda}^{(I(\lambda))}$ (No. 2). This proves the proposition.

Every simple module is primordial (VIII, p. 45), so every semisimple module is semiprimordial. Let M be a semisimple A-module, and let $\lambda \in \mathscr{S}$. The multiplicity of $\lambda$ in M is the primordial multiplicity $[M :\lambda ]$ of $\lambda$ in M defined in VIII, p. 34. Proposition 11 translates to the equality

(11) $[M :\lambda ] =$ dim$_{D_{\lambda}}$(Hom$_A(S_{\lambda},M)$).

More generally, if $((V_{\lambda})_{\lambda\in\mathscr{S}}, \alpha )$ is a description of M, then $[M :\lambda ]$ is equal to dim$_{D_{\lambda}}(V_{\lambda})$. By Proposition 6 of VIII, p. 68, we also have

(12) $[M :\lambda ] =$ dim$_{D_{\lambda}}$(Hom$_A(M,S_{\lambda})$)

when the multiplicity $[M :\lambda ]$ is finite. Semisimple A-modules M and $M'$ are isomorphic if and only if we have $[M :\lambda ] = [M':\lambda ]$ for every $\lambda \in \mathscr{S}$.

Let M be a semisimple A-module. There exists a cardinal $\mathbf{I}$ that has the following property: for every decomposition $M =\bigoplus_{i\in I}M_i$ of M into a direct sum of simple modules, the cardinal of I is equal to $\mathbf{I}$ (VIII, p. 34, Corollary 2). This cardinal is called the length of the semisimple A-module M and denoted by long$_A(M)$ or long(M). When M has finite length, this definition is compatible with that of II, §1, No. 10, p. 212, by Proposition 10.

The simple A-modules are the semisimple A-modules of length 1, and we have the formula

(13) long$_A(\bigoplus_{j\in J}M_j)=\sum_{j\in J}$ long$_A(M_j)$

for every family $(M_j)_{j\in J}$ of semisimple A-modules. By Proposition 11, we have

(14) long$_A(M) =\sum_{\lambda\in\mathscr{S}}$ dim$_{D_{\lambda}}$ Hom$_A(S_{\lambda},M)$.

By applying this formula to $M_{\lambda}$, we obtain $[M :\lambda ] =$ long$_A(M_{\lambda})$ for every $\lambda \in \mathscr{S}$.

When A is a field, the simple A-modules are the vector spaces of dimension 1; every A-module is then semisimple (II, §7, No. 1, p. 292, Theorem 1), and its length is simply its dimension as a vector space over A (II, §7, No. 2, p. 293) .