Representations of finite special linear groups in non-defining characteristic

TL;DR

This paper precisely determines the decomposition of certain representations of $GL_{n}(q)$ when restricted to $SL_{n}(q)$ and classifies complex representations of $SL_{n}(q)$ with irreducible reductions modulo $ell$, advancing the understanding of modular representation theory.

Contribution

It provides a detailed count of irreducible summands in cross characteristic representations of $GL_{n}(q)$ restricted to $SL_{n}(q)$ and classifies complex representations with irreducible modular reductions.

Findings

Exact number of irreducible summands determined.

Canonical labeling for irreducible $ell$-modular representations established.

First classification of complex representations with irreducible reductions modulo $ell$.

Abstract

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.