On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups

TL;DR

This paper investigates when certain quasisimple groups of Lie type can act irreducibly on tensor powers of a natural module of classical groups, shedding light on subgroup structures and representation irreducibility in cross-characteristic settings.

Contribution

It provides new criteria for irreducibility of group actions on tensor powers of modules, linking representation theory with subgroup classification in finite classical groups.

Findings

Identifies conditions for irreducibility of cross-characteristic representations

Establishes connections between irreducibility and maximal subgroup classification

Provides new insights into the structure of tensor product representations

Abstract

Let $W$ be a vector space over an algebraically closed field $k$. Let $H$ be a quasisimple group of Lie type of characteristic $p\ne {\rm char}(k)$ acting irreducibly on $W$. Suppose also that $G$ is a classical group with natural module $W$, chosen minimally with respect to containing the image of $H$ under the associated representation. We consider the question of when $H$ can act irreducibly on a $G$-constituent of $W^{\otimes e}$ and study its relationship to the maximal subgroup problem for finite classical groups.