Prime degree irreducible representations of simple algebraic groups and finite simple groups of Lie type

Abstract

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We also bound the number of such groups in terms of $r$. Apart from being notable in their own right, these results have a significant application in a computational version of the strong approximation theorem for finitely generated Zariski-dense subgroups of $SL_r(\mathbb{P})$, where $\mathbb{P}$ is a number field.