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The reductive subgroups of $F_4$

About this Title

David I. Stewart, New College, Oxford, OX1 3BN

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 223, Number 1049
ISBNs: 978-0-8218-8332-7 (print); 978-0-8218-9873-4 (online)
Published electronically: October 9, 2012
Keywords:Algebraic groups, exceptional groups, subgroup structure, positive characteristic, non-abelian cohomology, modular representation theory
MSC: Primary 20G07, 20G10, 20G41, 20G30, 18G50

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Overview
  • Chapter 3. General theory
  • Chapter 4. Reductive subgroups of $F_4$
  • Chapter 5. Appendices


Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p\geq 0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no proper parabolic subgroup of $G$; and $G$-reducible if it is in some proper parabolic of $G$. In this paper, we consider the case that $G=F_4(K)$.

We find all conjugacy classes of closed, connected, semisimple $G$-reducible subgroups $X$ of $G$. Thus we also find all non-$G$-completely reducible closed, connected, semisimple subgroups of $G$. When $X$ is closed, connected and simple of rank at least two, we find all conjugacy classes of $G$-irreducible subgroups $X$ of $G$. Together with the work of Amende classifying irreducible subgroups of type $A_1$ this gives a complete classification of the simple subgroups of $G$.

Amongst the classification of subgroups of $G=F_4(K)$ we find infinite varieties of subgroups $X$ of $G$ which are maximal amongst all reductive subgroups of $G$ but not maximal subgroups of $G$; thus they are not contained in any reductive maximal subgroup of $G$. The connected, semisimple subgroups contained in no maximal reductive subgroup of $G$ are of type $A_1$ when $p=3$ and of type $A_1^2$ or $A_1$ when $p=2$. Some of those which occur when $p=2$ act indecomposably on the 26-dimensional irreducible representation of $G$.

We also use this classification to find all subgroups of $G=F_4$ which are generated by short root elements of $G$, by utilising and extending the results of Leibeck and Seitz.

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