Non-G-completely reducible subgroups of the exceptional algebraic groups

TL;DR

This paper classifies all triples (X,G,p) where a simple, connected subgroup H of an exceptional algebraic group G over an algebraically closed field of characteristic p is not G-completely reducible, extending previous work on subgroup structure.

Contribution

It identifies all triples (X,G,p) with non-G-cr subgroups in exceptional algebraic groups, advancing understanding of subgroup reducibility in positive characteristic.

Findings

Complete classification of non-G-cr subgroups for exceptional groups

Identification of all triples (X,G,p) with such subgroups

Extension of previous subgroup reducibility results

Abstract

Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p>0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a parabolic subgroup P of G, then H is in a Levi subgroup of that parabolic. Building on work of Liebeck and Seitz, we find all triples (X,G,p) such that there exists a closed, connected, simple non-G-cr subgroup H<G with root system X.