Group Representation Theory

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 G is called irreducible if it lies in no proper parabolic subgroup of G 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 G -modules: for example, the conjugacy classes of irreducible connected subgroups are determined by their composition factors on the adjoint module of G G , with one exception. A result of Liebeck and Testerman shows that each irreducible connected subgroup X X of G G has only finitely many overgroups and hence the overgroups of X X form a lattice. We provide tables that give representatives of each conjugacy class of connected overgroups within this lattice structure. We use this to prove results concerning the subgroup structure of G G : for example, when the characteristic is 2 2 , there exists a maximal connected subgroup of G G containing a conjugate of every irreducible subgroup A 1 A_1 of G G .

Topological generation of exceptional algebraic groups

Simple irreducible subgroups of exceptional algebraic groups

Let Y be a simply connected, simple algebraic group of exceptional type, defined over an algebraically closed field k of characteristic p > 0.The main result describes all semisimple, closed connected subgroups of Y which act irreducibly on some rational kY module V, This extends work of Dynkin who obtained a similar classification for algebraically closed fields of characteristic 0. The main result has been combined with work of G. Seitz to obtain a classification of the maximal closed connected subgroups of the classical algebraic groups defined over k.AMS subject classification (1980).

Let G be a simple algebraic group of exceptional type over an algebraically closed field K of characteristic p. The purpose of this paper is to determine all maximal closed subgroups of G which act irreducibly on either the adjoint G-module or one of the well-known “minimal” modules of dimension 56, 27, 26− δp,3 or 7− δp,2 for G of type E7, E6, F4 or G2 respectively. (The adjective “minimal” here refers to the minimality of the dimension.) This greatly extends [21, Theorem 4] for exceptional groups.

Irreducible disconnected subgroups of exceptional algebraic groups

Jordan algebras, exceptional groups, and Bhargava composition

The involution fixity [Formula: see text] of a permutation group [Formula: see text] of degree [Formula: see text] is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if [Formula: see text] is the socle of such a group, then either [Formula: see text], or [Formula: see text] and [Formula: see text] is a Suzuki group in its natural [Formula: see text]-transitive action of degree [Formula: see text]. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with [Formula: see text]. This extends recent work of Liebeck and Shalev, who established the bound [Formula: see text] for every almost simple primitive group of degree [Formula: see text] with socle [Formula: see text] (with a prescribed list of exceptions). Finally, by combining our results with the Lang–Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle X ( q ) X(q) of Lie type in the natural characteristic. Our approach is to show that for sufficiently large q q (usually q > 9 q>9 suffices), X ( q ) X(q) is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.

This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E6, E7, and E8 as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra. Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms.

Let G be a finite exceptional group of Lie type acting transitively on a set O. For x in G, the fixed point ratio of x is the proportion of elements of O which are fixed by x. We obtain new bounds for such fixed point ratios. When a point-stabilizer is parabolic we use character theory; and in other cases, we use results on an analogous problem for algebraic groups in Lawther, Liebeck & Seitz, 2002. These give dimension bounds on fixed point spaces of elements of exceptional algebraic groups, which we apply by passing to finite groups via a Frobenius morphism.

In a number of cases the minimal polynomials of the images of unipotent elements of non-prime order in irreducible representations of the exceptional algebraic groups in good characteristics are found. It is proved that if p > 5 for a group of type E-8 and p > 3 for other exceptional algebraic groups, then for irreducible representations of these groups in characteristic p with large highest weights with respect to p, the degree of the minimal polynomial of the image of a unipotent element is equal to the order of this element.

Complete reducibility and subgroups of exceptional algebraic groups

Keywords: exceptional algebraic groups ; reductive groups ; connected subgroups ; Lie algebras ; labelled Dynkin ; diagrams ; conjugacy classes ; unipotent classes Reference CTG-ARTICLE-1999-001 Record created on 2008-12-16, modified on 2017-05-12

Irreducible subgroups of exceptional algebraic groups