The Irreducible Subgroups of Exceptional Algebraic Groups
The Irreducible Subgroups of Exceptional Algebraic Groups

Author: Adam R. Thomas

Publisher: American Mathematical Soc.

Published: 2021-06-18

Total Pages: 191

ISBN-13: 1470443376

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This paper is a contribution to the study of the subgroup structure of excep-tional 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 com-plete 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 sub-group 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 connected 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.

Irreducible Subgroups of Exceptional Algebraic Groups
Irreducible Subgroups of Exceptional Algebraic Groups

Author: Donna M. Testerman

Publisher: American Mathematical Soc.

Published: 1988

Total Pages: 198

ISBN-13: 0821824538

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Let [italic]Y be a simply-connected, simple algebraic group of exceptional type, defined over an algebraically closed field [italic]k of prime characteristic [italic]p > 0. The main result describes all semisimple, closed connected subgroups of [italic]Y which act irreducibly on some rational [italic]k[italic]Y module [italic]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 [italic]k.

On Non-Generic Finite Subgroups of Exceptional Algebraic Groups
On Non-Generic Finite Subgroups of Exceptional Algebraic Groups

Author: Alastair J. Litterick

Publisher: American Mathematical Soc.

Published: 2018-05-29

Total Pages: 168

ISBN-13: 1470428377

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The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic.

Reductive Subgroups of Exceptional Algebraic Groups
Reductive Subgroups of Exceptional Algebraic Groups

Author: Martin W. Liebeck

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 122

ISBN-13: 0821804618

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The theory of simple algebraic groups is important in many areas of mathematics. The authors of this book investigate the subgroups of certain types of simple algebraic groups and obtain a complete description of all those subgroups which are themselves simple. This description is particularly useful in understanding centralizers of subgroups and restrictions of representations.

The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups
The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups

Author: Martin W. Liebeck

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 242

ISBN-13: 0821834827

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Intends to complete the determination of the maximal subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields. This title follows work of Dynkin, who solved the problem in characteristic zero, and Seitz who did likewise over fields whose characteristic is not too small.

The Maximal Subgroups of Classical Algebraic Groups
The Maximal Subgroups of Classical Algebraic Groups

Author: Gary M. Seitz

Publisher: American Mathematical Soc.

Published: 1987

Total Pages: 294

ISBN-13: 0821824279

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Let [italic]V be a finite dimensional vector space over an algebraically closed field of characteristic p [greater than] 0 and let G = SL([italic]V), Sp([italic]V), or SO([italic]V). The main result describes all closed, connected, overgroups of [italic]X in SL([italic]V), assuming [italic]X is a closed, connected, irreducible subgroup of G.

Linear Algebraic Groups and Finite Groups of Lie Type
Linear Algebraic Groups and Finite Groups of Lie Type

Author: Gunter Malle

Publisher: Cambridge University Press

Published: 2011-09-08

Total Pages: 324

ISBN-13: 113949953X

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Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

Irreducible Geometric Subgroups of Classical Algebraic Groups
Irreducible Geometric Subgroups of Classical Algebraic Groups

Author: Timothy C. Burness,

Publisher: American Mathematical Soc.

Published: 2016-01-25

Total Pages: 100

ISBN-13: 1470414945

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Let be a simple classical algebraic group over an algebraically closed field of characteristic with natural module . Let be a closed subgroup of and let be a non-trivial irreducible tensor-indecomposable -restricted rational -module such that the restriction of to is irreducible. In this paper the authors classify the triples of this form, where is a disconnected maximal positive-dimensional closed subgroup of preserving a natural geometric structure on .

Irreducible Almost Simple Subgroups of Classical Algebraic Groups
Irreducible Almost Simple Subgroups of Classical Algebraic Groups

Author: Timothy C. Burness

Publisher: American Mathematical Soc.

Published: 2015-06-26

Total Pages: 122

ISBN-13: 147041046X

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Let be a simple classical algebraic group over an algebraically closed field of characteristic with natural module . Let be a closed subgroup of and let be a nontrivial -restricted irreducible tensor indecomposable rational -module such that the restriction of to is irreducible. In this paper the authors classify the triples of this form, where and is a disconnected almost simple positive-dimensional closed subgroup of acting irreducibly on . Moreover, by combining this result with earlier work, they complete the classification of the irreducible triples where is a simple algebraic group over , and is a maximal closed subgroup of positive dimension.