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Publication# Irreducibility of disconnected subgroups of exceptional algebraic groups

Abstract

This dissertation is concerned with the study of irreducible embeddings of simple algebraic groups of exceptional type. It is motivated by the role of such embeddings in the study of positive dimensional closed subgroups of classical algebraic groups. The classification of the maximal closed connected subgroups of simple algebraic groups was carried out by E. B. Dynkin, G. M. Seitz and D. M. Testerman. Their analysis for the classical groups was based primarily on a striking result: if G is a simple algebraic group and ø : G → SL(V ) is a tensor indecomposable irreducible rational representation then, with specified exceptions, the image of G is maximal among closed connected subgroups of one of the classical groups SL(V), Sp(V ) or SO(V ). In the case of closed, not necessarily connected, subgroups of the classical groups, one is interested in considering irreducible embeddings of simple algebraic groups and their automorphism groups: given a simple algebraic group Y defined over an algebraically closed field K, one is led to study the embeddings G < Aut(Y ), where G and Aut(Y ) are closed subgroups of SL(V ) and V is an irreducible rational KY -module on which G acts irreducibly. A partial analysis of such embeddings in the case of classical algebraic groups Y was carried out by B. Ford. We purpose to classify all triples (G, Y, V ) where Y is a simple algebraic group of exceptional type, defined over an algebraically closed field K of characteristic p > 0, G is a closed non-connected positive dimensional subgroup of Y and V is a nontrivial irreducible rational KY -module such that V|G is irreducible. We obtain a precise description of such triples (G, Y, V ).

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Related publications (11)

Related concepts (18)

Related MOOCs (9)

Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted H ≤ G, read as "H is a subgroup of G". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G).

Simple Lie group

In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension.

Analysis

Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent development. The word comes from the Ancient Greek ἀνάλυσις (analysis, "a breaking-up" or "an untying;" from ana- "up, throughout" and lysis "a loosening"). From it also comes the word's plural, analyses.

Algebra (part 1)

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

Algebra (part 1)

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

Algebra (part 2)

Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$. In this thesis, we investigate closed connected reductive subgroups $X < G$ that contain a given distinguished unipotent element $u$ of $G$. Our main result is the classification of all such $X$ that are maximal among the closed connected subgroups of $G$.
When $G$ is simple of exceptional type, the result is easily read from the tables computed by Lawther (J. Algebra, 2009). Our focus is then on the case where $G$ is simple of classical type, say $G = \operatorname{SL}(V)$, $G = \operatorname{Sp}(V)$, or $G = \operatorname{SO}(V)$. We begin by considering the maximal closed connected subgroups $X$ of $G$ which belong to one of the families of the so-called \emph{geometric subgroups}. Here the only difficult case is the one where $X$ is the stabilizer of a tensor decomposition of $V$. For $p = 2$ and $X = \operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)$, we solve the problem with explicit calculations; for the other tensor product subgroups we apply a result of Barry (Comm. Algebra, 2015).
After the geometric subgroups, the maximal closed connected subgroups that remain are the $X < G$ such that $X$ is simple and $V$ is an irreducible and tensor indecomposable $X$-module. The bulk of this thesis is concerned with this case. We determine all triples $(X, u, \varphi)$ where $X$ is a simple algebraic group, $u \in X$ is a unipotent element, and $\varphi: X \rightarrow G$ is a rational irreducible representation such that $\varphi(u)$ is a distinguished unipotent element of $G$. When $p = 0$, this was done in previous work by Liebeck, Seitz and Testerman (Pac. J. Math, 2015).
In the final chapter of the thesis, we consider the more general problem of finding all connected reductive subgroups $X$ of $G$ that contain a distinguished unipotent element $u$ of $G$. This leads us to consider connected reductive overgroups $X$ of $u$ which are contained in some proper parabolic subgroup of $G$. Testerman and Zalesski (Proc. Am. Math. Soc, 2013) have shown that when $u$ is a regular unipotent element of $G$, no such $X$ exists. We give several examples which show that their result does not generalize to distinguished unipotent elements. As an extension of the Testerman-Zalesski result, we show that except for two known examples which occur in the case where $(G, p) = (C_2, 2)$, a connected reductive overgroup of a distinguished unipotent element of order $p$ cannot be contained in a proper parabolic subgroup of $G$.

Let G be a simple linear algebraic group over an algebraically closed field K of characteristic p≥ 0, let H be a proper closed subgroup of G and let V be a nontrivial finite dimensional irreducible rational KG-module. We say that (G,H, V) is an irreducible triple if V is irreducible as a KH-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.

Let K be an algebraically closed field of characteristic $p\geq0$ and let $Y=SPin_{2n+1}(K) (n\geq3)$ be a simply connected simple algebraic group of type $B_n$ over $K$. Also let $X$ be the subgroup of type $D_n$, embedded in $Y$ in the usual way, as the derived subgroup of the stabilizer of a non-singular one-dimensional subspace of the natural module for $Y$. In this paper, we give a complete set of isomorphism classes of finite-dimensional, irreducible, rational $KY$-modules on which $X$ acts with exactly two composition factors, completing the work of Ford in [12].

2017