The Irreducible Subgroups of Exceptional Algebraic Groups

The Irreducible Subgroups of Exceptional Algebraic Groups
Title The Irreducible Subgroups of Exceptional Algebraic Groups PDF eBook
Author Adam R. Thomas
Publisher American Mathematical Soc.
Pages 191
Release 2021-06-18
Genre Education
ISBN 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
Title Irreducible Subgroups of Exceptional Algebraic Groups PDF eBook
Author Donna M. Testerman
Publisher American Mathematical Soc.
Pages 198
Release 1988
Genre Embeddings
ISBN 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
Title On Non-Generic Finite Subgroups of Exceptional Algebraic Groups PDF eBook
Author Alastair J. Litterick
Publisher American Mathematical Soc.
Pages 168
Release 2018-05-29
Genre Mathematics
ISBN 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
Title Reductive Subgroups of Exceptional Algebraic Groups PDF eBook
Author Martin W. Liebeck
Publisher American Mathematical Soc.
Pages 122
Release 1996
Genre Mathematics
ISBN 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.

$A_1$ Subgroups of Exceptional Algebraic Groups

$A_1$ Subgroups of Exceptional Algebraic Groups
Title $A_1$ Subgroups of Exceptional Algebraic Groups PDF eBook
Author Ross Lawther
Publisher American Mathematical Soc.
Pages 146
Release 1999
Genre Mathematics
ISBN 0821819666

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This book is intended for graduate students and research mathematicians interested in group theory and genralizations

The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups

The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups
Title The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups PDF eBook
Author Martin W. Liebeck
Publisher American Mathematical Soc.
Pages 242
Release 2004
Genre Mathematics
ISBN 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
Title The Maximal Subgroups of Classical Algebraic Groups PDF eBook
Author Gary M. Seitz
Publisher American Mathematical Soc.
Pages 294
Release 1987
Genre Linear algebraic groups
ISBN 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.