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