Reductive Subgroups Of Exceptional Algebraic Groups

Author : Martin W. Liebeck,Gary M. Seitz
Publisher : American Mathematical Soc.
Page : 111 pages
File Size : 53,6 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804612

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Reductive Subgroups of Exceptional Algebraic Groups by Martin W. Liebeck,Gary M. Seitz Pdf

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.

Author : Gary M. Seitz
Publisher : American Mathematical Soc.
Page : 197 pages
File Size : 45,8 Mb
Release : 1991
Category : Mathematics
ISBN : 9780821825044

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Maximal Subgroups of Exceptional Algebraic Groups by Gary M. Seitz Pdf

The goal of this book is the determination of the maximal closed connected subgroups of the simple algebraic groups of exceptional type. The main result recovers the results of Dynkin and extends them to cover the case of algebraic groups over algebraically closed fields of positive characteristic.

Author : Donna M. Testerman
Publisher : American Mathematical Soc.
Page : 198 pages
File Size : 54,8 Mb
Release : 1988
Category : Embeddings
ISBN : 9780821824535

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Irreducible Subgroups of Exceptional Algebraic Groups by Donna M. Testerman Pdf

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.

Author : R.W. Carter,J. Saxl
Publisher : Springer Science & Business Media
Page : 396 pages
File Size : 48,5 Mb
Release : 1998-08-31
Category : Mathematics
ISBN : 0792352920

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Algebraic Groups and Their Representations by R.W. Carter,J. Saxl Pdf

This volume contains articles by 20 leading workers in the field of algebraic groups and related finite groups. Articles on representation theory are written by Andersen on tilting modules, Carter on canonical bases, Cline, Parshall and Scott on endomorphism algebras, James and Kleshchev on the symmetric group, Littelmann on the path model, Lusztig on homology bases, McNinch on semisimplicity in prime characteristic, Robinson on block theory, Scott on Lusztig's character formula, and Tanisaki on highest weight modules. Articles on subgroup structure are written by Seitz and Brundan on double cosets, Liebeck on exceptional groups, Saxl on subgroups containing special elements, and Guralnick on applications of subgroup structure. Steinberg gives a new, short proof of the isomorphism and isogeny theorems for reductive groups. Aschbacher discusses the classification of quasithin groups and Borovik the classification of groups of finite Morley rank. Audience: The book contains accounts of many recent advances and will interest research workers and students in the theory of algebraic groups and related areas of mathematics.

Author : Roger W. Carter,Meinolf Geck
Publisher : Cambridge University Press
Page : 203 pages
File Size : 46,5 Mb
Release : 1998-09-03
Category : Mathematics
ISBN : 9780521643252

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Representations of Reductive Groups by Roger W. Carter,Meinolf Geck Pdf

This volume provides a very accessible introduction to the representation theory of reductive algebraic groups.

Author : Adam R. Thomas
Publisher : American Mathematical Soc.
Page : 191 pages
File Size : 54,7 Mb
Release : 2021-06-18
Category : Education
ISBN : 9781470443375

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The Irreducible Subgroups of Exceptional Algebraic Groups by Adam R. Thomas Pdf

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.

Author : Alastair J. Litterick
Publisher : American Mathematical Soc.
Page : 156 pages
File Size : 49,6 Mb
Release : 2018-05-29
Category : Affine algebraic groups
ISBN : 9781470428372

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On Non-Generic Finite Subgroups of Exceptional Algebraic Groups by Alastair J. Litterick Pdf

Author : Ross Lawther,Donna M. Testerman
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 52,9 Mb
Release : 1999
Category : Lie algebras
ISBN : 9780821819661

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$A_1$ Subgroups of Exceptional Algebraic Groups by Ross Lawther,Donna M. Testerman Pdf

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

Author : Martin W. Liebeck,Gary M. Seitz
Publisher : American Mathematical Soc.
Page : 227 pages
File Size : 47,8 Mb
Release : 2004
Category : Mathematics
ISBN : 9780821834824

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The Maximal Subgroups of Positive Dimension in Exceptional Algebraic Groups by Martin W. Liebeck,Gary M. Seitz Pdf

In this paper, we complete the determination of the maximal subgroups of positive dimension in simple algebraic groups of exceptional type over algebraically closed fields. This follows work of Dynkin, who solved the problem in characteristic zero, and Seitz who did likewise over fields whose characteristic is not too small. A number of consequences are obtained. It follows from the main theorem that a simple algebraic group over an algebraically closed field has only finitely many conjugacy classes of maximal subgroups of positive dimension. It also follows that the maximal subgroups of sufficiently large order in finite exceptional groups of Lie type are known.

Author : David I. Stewart
Publisher : American Mathematical Soc.
Page : 88 pages
File Size : 45,6 Mb
Release : 2013-04-22
Category : Mathematics
ISBN : 9780821883327

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The Reductive Subgroups of $F_4$ by David I. Stewart Pdf

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, the author considers the case that $G=F_4(K)$. The author finds all conjugacy classes of closed, connected, semisimple $G$-reducible subgroups $X$ of $G$. Thus he also finds all non-$G$-completely reducible closed, connected, semisimple subgroups of $G$. When $X$ is closed, connected and simple of rank at least two, he finds 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 $G$. The author also uses 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 Liebeck and Seitz.

Author : Brian Conrad,Gopal Prasad
Publisher : Princeton University Press
Page : 256 pages
File Size : 53,9 Mb
Release : 2015-11-10
Category : Mathematics
ISBN : 9781400874026

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Classification of Pseudo-reductive Groups (AM-191) by Brian Conrad,Gopal Prasad Pdf

In the earlier monograph Pseudo-reductive Groups, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudo-reductive groups. In this new book, Classification of Pseudo-reductive Groups, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the automorphism schemes of these groups. The book also gives a Tits-Witt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudo-split forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in Classification of Pseudo-reductive Groups will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic.

Author : David A. Craven
Publisher : American Mathematical Society
Page : 168 pages
File Size : 41,9 Mb
Release : 2022-04-08
Category : Mathematics
ISBN : 9781470451196

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Maximal $textrm {PSL}_2$ Subgroups of Exceptional Groups of Lie Type by David A. Craven Pdf

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Author : Brian Conrad,Ofer Gabber,Gopal Prasad
Publisher : Cambridge University Press
Page : 555 pages
File Size : 53,8 Mb
Release : 2010-07-29
Category : Mathematics
ISBN : 9781139490344

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Pseudo-reductive Groups by Brian Conrad,Ofer Gabber,Gopal Prasad Pdf

Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This self-contained monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudo-parabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book as it develops tools likely to be used in tackling other problems.

Author : Gary M. Seitz
Publisher : American Mathematical Soc.
Page : 294 pages
File Size : 41,7 Mb
Release : 1987
Category : Linear algebraic groups
ISBN : 9780821824276

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The Maximal Subgroups of Classical Algebraic Groups by Gary M. Seitz Pdf

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.

Author : Ross Lawther
Publisher : American Mathematical Society(RI)
Page : 146 pages
File Size : 40,6 Mb
Release : 2014-09-11
Category : Lie algebras
ISBN : 1470402653

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A Subgroups of Exceptional Algebraic Groups by Ross Lawther Pdf

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