Representations of finite special linear groups in non-defining characteristic

We determine precisely the number of irreducible summands of an irreducible cross characteristic representation of $G{L}_n(q)$ on restriction to $S{L}_n(q)$. Combined with a recent result of C. Bonnafe, this yields a canonical labeling for irreducible $\mathrm{\ell }$-modular representations of $S{L}_n(q)$, where $(\mathrm{\ell },q)=1$. As an application, we classify for the first time complex representations of $S{L}_n(q)$ whose reductions modulo $\mathrm{\ell }$ are irreducible.