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Modular representations of some finite simple groups

Ibrahim A. I. Suleiman

Modular representations of some finite simple groups

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Published by Universityof Birmingham in Birmingham .
Written in English


Edition Notes

Thesis (Ph.D.)-University of Birmingham, School of Mathematics and Statistics.

Statementby Ibrahim A.I. Suleiman.
ID Numbers
Open LibraryOL13891286M

Modular Representations of Finite Groups (Pure and Applied Mathematics (Academic Press), Volume 73) | B. M. Puttaswamaiah, John D. Dixon | download | B–OK. Download books for free. Find books. phisms. In some cases the space obtained by continuation is compact. For example, this is the cas e for all modular abelian groups and, in particular, for Siegel' s modular group. For these latter, the continuation described in We sometimes require only invariance under a . springer, Representation theory studies maps from groups into the general linear group of a finite-dimensional vector space. For finite groups the theory comes in two distinct flavours. In the 'semisimple case' (for example over the field of complex numbers) one can use character theory to completely understand the representations. This by far is not sufficient when the characteristic of the.


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Modular representations of some finite simple groups by Ibrahim A. I. Suleiman Download PDF EPUB FB2

Finite groups of Lie type encompass most of the finite simple groups. Their representations and characters have been studied intensively for half a Modular representations of some finite simple groups book, though some key problems remain unsolved.

This is the first comprehensive treatment of the representation theory of finite groups of Lie type over a field of the defining prime characteristic. Finite groups of Lie type encompass most of the finite simple groups.

Their representations and characters have been studied intensively for half a century, though some key problems Modular representations of some finite simple groups book unsolved. Modular representations of some finite simple groups book is the first comprehensive treatment of the representation theory of finite groups of Modular representations of some finite simple groups book type over a field of the defining prime : $ Representations of Finite Groups provides an account of the fundamentals of ordinary and modular representations.

This book discusses the fundamental theory of complex representations of finite groups. Organized into five chapters, this book begins with an overview. Modular representation theory is a branch of mathematics, Modular representations of some finite simple groups book that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory [citation.

This cohomological approach carries over to the context of algebraic groups in characteristic p. In essence, this is where the book Modular Representations of Finite Groups of Lie Type begins.

After a brief introduction, Humphreys guides the reader through the some of the major theorems and conjectures in the subject. Modular Representations of Finite Groups of Lie Type. Summary: The first comprehensive treatment of the representation theory of finite groups of Lie type over a field of the defining prime characteristic.

We determine for all simple simply connected reductive linear algebraic groups defined over a finite field all irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining.

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. Group theory is central to many areas of pure and applied mathematics and the classification.

The modular representation theory of finite groups has its origins in the work of Richard Brauer. In this survey article we first discuss the work being done on some outstanding conjectures in the Author: Bhama Srinivasan. thereby giving representations of the group on the homology groups of the space.

If there is torsion in the homology these representations require something other than ordinary character theory to be understood. This book is written for students who are studying nite group representation theory beyond the level of a rst course in abstract Size: 1MB.

Get this from a library. Local representation theory: modular representations as an introduction to the local representation theory of finite groups. [J L Alperin]. The Mathematical Sciences Research Institute (MSRI), founded inis an independent nonprofit mathematical research institution whose funding sources include the National Science Foundation, foundations, corporations, and more than 90 universities and institutions.

The Institute is located at 17 Gauss Way, on the University of California, Berkeley campus, close to Grizzly Peak, on the. In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.

The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics.

Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable Cited by: 8. Let be a prime that does not be the order of modulo and e that mod so and therefore contains the -th roots of unity in.

Proposition The ring of algebraic integers in is. In other words, the integral closure of in do not really need this somewhat deep result, and will denote by the integral closure of in.

Proof. Modular Representation theory, the study of representations of finite groups over a field of positive characteristic, has in particular been used in the classification of finite simple groups, and itself finds applications in a variety of areas of mathematics. The modular representation theory of finite groups has its origins in the work of Richard Brauer.

In this survey article, we first discuss the work being done on some outstanding conjectures in the theory. We then describe work done in the eighties and nineties on modular representations in non-defining characteristic of finite groups of Lie : Bhama Srinivasan.

These contributors include some of the most active researchers in areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, and more. The reader is referred to [11] and [13] for background on representations of symmetric groups, and [14] or [17] for representations of finite groups.

Fix r ∈ N. Fix r ∈ N. Representation Theory of Finite Groups is a five chapter text that covers the standard material of representation theory. This book starts with an overview of the basic concepts of the subject, including group characters, representation modules, and the rectangular representation.

Linear codes obtained from 2-modular representations of some finite simple groups. View/ Open. Thesis. (Mb) The approach is modular representation theoretic and based on a study of maximal submodules of permutation modules F defined by the action of a finite group G on G-sets = G=Gx.

In this thesis we have surveyed some known. Alternatively: You can't always lift a modular representation to characteristic zero, but it turns out you can always lift modular representations that are projective (as $\bar{\mathbb{F}}_pG$ - modules) to characteristic zero.

The generic finite simple group is a finite group of Lie type. Each such group Gis described via a representation as a linear group, say G≤GL(V) for some finite-dimensional vector space Vover some finite field F. Thus Ghas a characteristic which is a prime: The characteristic pof F.

Similarly the Lie. Representations of Finite Groups provides an account of the fundamentals of ordinary and modular representations. This book discusses the fundamental theory of complex representations of finite groups. Organized into five chapters, this book begins with an overview of the basic facts about rings and Edition: 1.

THE 5-MODULAR REPRESENTATIONS OF THE TITS SIMPLE GROUP construct the Green correspondents and sources, and second to work out their socle series. The reader is referred to the book of Landrock [5] and to the paper by Schneider [9] for the necessary background from representation theory and a detailed description of the algorithms of §4.

This is a great book on modular representation theory, focusing on the basics of the theory and how the (projective, indecomposable, simple, etc.) modules over a local subgroup correspond to similar modules over the whole group.5/5.

What is modular representation for finite groups. I tried to find a book to understanding that but I could not find a good one. Are there any useful references.

Some main problems of modular representation theory: Describe the irreducible modular representations, e.g. their degrees Describe the blocks Find the decomposition matrix D Global to local: Describe information on the block B by ”local information”, i.e.

from blocks of File Size: KB. Modular representation theory played a key role in the classification of finite simple groups. More recently, beginning with work of Lascoux, Leclerc and Thibon, deep connections have been found with the representation theory of quantum groups and modular representation theory, for example of the symmetric group.

Hi, a very elementary written book is Local Representation Theory by second book on finite groups by Huppert has also a big part about modular representation theory.(you should read the first book too,with an long introduction to representation theory in the semisimple case) a more advanced book is that of feit and a recent () book is "Representations of Groups: A Computational.

We give some general reductions towards the determination of the character of the simple module. Its highest weight is identified and the problem is reduced to the case of a prime field. The reduced problem can be approached through the representation theory of algebraic groups and the methods are illustrated for some by: 3.

A review of some classes of finite groups Sylow's theorem Linear representations of supersolvable groups 9 Artin 's theorem The ring R(G) Statement of Artin's theorem First proof Second proof of (i) ~ (ii) 10 A theorem of Brauer p-regular elements; p-elementary subgroups.

Abstract. Since all finite simple groups have been classified [], it is a natural question whether the major conjectures in modular representation theory are consequences of this important and deep classification this article, a survey is given about the progress which has been achieved on the famous open conjectures of J.

Alperin [] and R. Brauer [] during the last by: 5. I study the structure and representations of linear algebraic groups, and most of the questions he considers involve the "modular setting" where the coefficient field contains some finite field - thus, the characteristic of the coefficient field is a prime number p>0.

Modular Representation Theory of finite Groups comprises this second situation. Many additional tools are needed for this case. To mention some, there is the systematic use of Grothendieck groups leading to the Cartan matrix and the decomposition matrix of the group as well as Green's direct analysis of indecomposable : Theory of Finite Simple Groups This book provides the first representation theoretic and algorithmic approach to Modular characters of finite groups Blocks of defect zero 5 Permutation representations Permutation groups.

This book is an outgrowth of a Research Symposium on the Modular Representation Theory of Finite Groups, held at the University of Virginia in May The main themes of this symposium were representations of groups of Lie type in nondefining (or cross) characteristic, and recent developments in block theory.

Series of lectures were given by M. Geck, A. Kleshchev and R. Rouquier, and their. Modular Representation Theory of finite Groups comprises this second situation. Many additional tools are needed for this case. To mention some, there is the systematic use of Grothendieck groups leading to the Cartan matrix and the decomposition matrix of the group as well as Green's direct analysis of indecomposable representations.

Originally written inthis book remains a classical source on representations and characters of finite and compact groups. The book starts with necessary information about matrices, algebras, and groups.

Then the author proceeds to representations of finite groups. Stratifying modular representations of finite groups. Pages from Volume (), Issue 3 by David J. Benson, localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules.

@book {Bensona, MRKEY = {},Cited by:. History. During pdf twentieth century, mathematicians investigated some aspects pdf the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups.

As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known.The Modular Representation Theory of Finite Groups This dissertation will develop modular representation theory, starting from Brauer’s Three Main Theorems.

We will consider the Green Correspondence, then use G-algebras to unite the block-theoretic and module-theoretic approaches somewhat. We then consider someFile Size: KB.1 0 Introduction Let Gbe a finite group, let K be ebook field, and let V ebook a finite-dimensional vector space over by GL(V) the group of invertible linear transformations from V to itself.A group homomorphism ρ: G→ GL(V) is called a linear K-representation of Gin V (or just a representation of Gfor short).

One gains information about the structure of Gby studying the totality of.