The Irreducible Subgroups of Exceptional Algebraic Groups

TL;DR

This paper completes the classification of irreducible connected subgroups of exceptional algebraic groups over algebraically closed fields, providing explicit representatives and analyzing their conjugacy classes, representations, and overgroups.

Contribution

It provides a complete classification of irreducible connected subgroups of exceptional algebraic groups, including explicit representatives and overgroup lattice structures.

Findings

Conjugacy classes of irreducible subgroups are determined by their composition factors.

Finiteness of overgroups for each irreducible subgroup is established.

Existence of maximal connected subgroups containing conjugates of all irreducible A1 subgroups in characteristic 2.

Abstract

This paper is a contribution to the study of the subgroup structure of exceptional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group $G$ is called irreducible if it lies in no proper parabolic subgroup of $G$. In this paper we complete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning the representations of such subgroups on various $G$-modules: for example, the conjugacy classes of irreducible connected subgroups are determined by their composition factors on the adjoint module of $G$, with one exception. A result of Liebeck and Testerman shows that each irreducible connected subgroup $X$ of $G$ has only finitely many overgroups and hence the overgroups of $X$ form a lattice. We provide tables that give representatives of each conjugacy class of overgroups within this lattice structure. We use this to prove results concerning the subgroup structure of $G$: for example, when the characteristic is 2, there exists a maximal connected subgroup of $G$ containing a conjugate of every irreducible subgroup $A_1$ of $G$.