The Reductive Subgroups Of F4

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The Reductive Subgroups Of F4

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Author : David I. Stewart
language : en
Publisher:
Release Date : 2013

The Reductive Subgroups Of F4 written by David I. Stewart and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Categories categories.

Let G=G(K) be a simple algebraic group defined over an algebraically closed field K of characteristic p ≥ 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, we consider the case that G = F4(K). We find all conjugacy classes of closed, connected, semisimple G-reducible subgroups X of G. Thus we also find all non-G-completely reducible closed, connected, semisimple subgroups of G. When X is closed, connected and simple of rank at least two, we find all conjugacy classes of G-irreducible subgroups X of G. Together with the work of Amende classifying irreducible subgroups of type A1 this gives a complete classification of the simple subgroups of G. Amongst the classification of subgroups G=F4(K) we find infinite varieties of subgroups X of G which are maximal amongst all reductive subgroups of G but not maximal subgroups of G; thus they are not contained in any reductive maximal subgroup of G. The connected, semisimple subgroups contained in no maximal reductive subgroup of G are of type A1 when p=3 and of type A21 or A1 when p = 2. Some of those which occur when p=2 act indecomposably on the 26-dimensional irreducible representation of G. We also use this classification to find all subgroups of G=F4 which are generated by short root elements of G, by utilising and extending the results of Leibeck and Seitz.

The Reductive Subgroups Of F 4

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Author : David I. Stewart
language : en
Publisher: American Mathematical Soc.
Release Date : 2013-04-22

The Reductive Subgroups Of F 4 written by David I. Stewart and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-22 with Mathematics categories.

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.

The Irreducible Subgroups Of Exceptional Algebraic Groups

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Author : Adam R. Thomas
language : en
Publisher: American Mathematical Soc.
Release Date : 2021-06-18

The Irreducible Subgroups Of Exceptional Algebraic Groups written by Adam R. Thomas and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-18 with Education categories.

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.

Reductive Subgroups Of Exceptional Algebraic Groups

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Author : Martin W. Liebeck
language : en
Publisher: American Mathematical Soc.
Release Date : 1996

Reductive Subgroups Of Exceptional Algebraic Groups written by Martin W. Liebeck and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Mathematics categories.

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.

Maximal Textrm Psl 2 Subgroups Of Exceptional Groups Of Lie Type

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Author : David A. Craven
language : en
Publisher: American Mathematical Society
Release Date : 2022-04-08

Maximal Textrm Psl 2 Subgroups Of Exceptional Groups Of Lie Type written by David A. Craven and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-08 with Mathematics categories.

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On Medium Rank Lie Primitive And Maximal Subgroups Of Exceptional Groups Of Lie Type

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Author : David A. Craven
language : en
Publisher: American Mathematical Society
Release Date : 2023-09-15

On Medium Rank Lie Primitive And Maximal Subgroups Of Exceptional Groups Of Lie Type written by David A. Craven and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-09-15 with Mathematics categories.

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The Maximal Subgroups Of Positive Dimension In Exceptional Algebraic Groups

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Author : Martin W. Liebeck
language : en
Publisher: American Mathematical Soc.
Release Date : 2004

The Maximal Subgroups Of Positive Dimension In Exceptional Algebraic Groups written by Martin W. Liebeck and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.

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.

Kuznetsov S Trace Formula And The Hecke Eigenvalues Of Maass Forms

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Author : Andrew Knightly
language : en
Publisher: American Mathematical Soc.
Release Date : 2013-06-28

Kuznetsov S Trace Formula And The Hecke Eigenvalues Of Maass Forms written by Andrew Knightly and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-28 with Mathematics categories.

The authors give an adelic treatment of the Kuznetsov trace formula as a relative trace formula on $\operatorname{GL}(2)$ over $\mathbf{Q}$. The result is a variant which incorporates a Hecke eigenvalue in addition to two Fourier coefficients on the spectral side. The authors include a proof of a Weil bound for the generalized twisted Kloosterman sums which arise on the geometric side. As an application, they show that the Hecke eigenvalues of Maass forms at a fixed prime, when weighted as in the Kuznetsov formula, become equidistributed relative to the Sato-Tate measure in the limit as the level goes to infinity.

Cohomology For Quantum Groups Via The Geometry Of The Nullcone

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Author : Christopher P. Bendel
language : en
Publisher: American Mathematical Soc.
Release Date : 2014-04-07

Cohomology For Quantum Groups Via The Geometry Of The Nullcone written by Christopher P. Bendel and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-04-07 with Mathematics categories.

In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p ) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible G -modules stipulates that p=h. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra H (u ? ,C) of the small quantum group.

On Non Generic Finite Subgroups Of Exceptional Algebraic Groups

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Author : Alastair J. Litterick
language : en
Publisher: American Mathematical Soc.
Release Date : 2018-05-29

On Non Generic Finite Subgroups Of Exceptional Algebraic Groups written by Alastair J. Litterick and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-29 with Mathematics categories.

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.