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

TL;DR

This paper establishes bounds on the size and number of finite simple groups of Lie type with prime degree irreducible representations over finite fields, with implications for computational number theory.

Contribution

It provides bounds on the order and quantity of such groups based on prime degree, advancing understanding of their structure and applications.

Findings

Finite quasisimple groups of Lie type with prime degree irreducible representations have bounded order.

The number of such groups is bounded in terms of the prime degree.

Results have applications in computational aspects of the strong approximation theorem.

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.