Thomas, Adam (2017) The irreducible subgroups of exceptional algebraic groups. Memoirs of the American Mathematical Society . (In Press)

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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 A1 of G.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Algebraic fields, Group theory
Journal or Publication Title:	Memoirs of the American Mathematical Society
Publisher:	American Mathematical Society
ISSN:	0065-9266
Official Date:	1 October 2017
Dates:	Date Event 1 October 2017 Accepted
Date of first compliant deposit:	15 October 2019
Status:	Peer Reviewed
Publication Status:	In Press
Publisher Statement:	“First published in Memoirs of the American Mathematical Society in [volume/issue number and year], published by the American Mathematical Society,” and the copyright notice in proper form must be placed on all copies.
Access rights to Published version:	Restricted or Subscription Access
Copyright Holders:	American Mathematical Society
Open Access Version:	ArXiv

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