Group Representation Theory Modular Representation Theory

Author : Peter Schneider
Publisher : Springer Science & Business Media
ISBN 13 : 1447148320
Total Pages : 183 pages
Book Rating : 4.26/5 ( download)

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Download or read book Modular Representation Theory of Finite Groups written by Peter Schneider and published by Springer Science & Business Media. This book was released on 2012-11-27 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 field divides the order of the group. 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. There is also the strategy of writing the category of all representations as the direct product of certain subcategories, the so-called 'blocks' of the group. Brauer's work then establishes correspondences between the blocks of the original group and blocks of certain subgroups the philosophy being that one is thereby reduced to a simpler situation. In particular, one can measure how nonsemisimple a category a block is by the size and structure of its so-called 'defect group'. All these concepts are made explicit for the example of the special linear group of two-by-two matrices over a finite prime field. Although the presentation is strongly biased towards the module theoretic point of view an attempt is made to strike a certain balance by also showing the reader the group theoretic approach. In particular, in the case of defect groups a detailed proof of the equivalence of the two approaches is given. This book aims to familiarize students at the masters level with the basic results, tools, and techniques of a beautiful and important algebraic theory. Some basic algebra together with the semisimple case are assumed to be known, although all facts to be used are restated (without proofs) in the text. Otherwise the book is entirely self-contained.

Author : D. Benson
Publisher : Springer Science & Business Media
ISBN 13 : 3540133895
Total Pages : 246 pages
Book Rating : 4.96/5 ( download)

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Download or read book Modular Representation Theory written by D. Benson and published by Springer Science & Business Media. This book was released on 1984 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: This reprint of a 1983 Yale graduate course makes results in modular representation theory accessible to an audience ranging from second-year graduate students to established mathematicians. Following a review of background material, the lectures examine three closely connected topics in modular representation theory of finite groups: representations rings; almost split sequences and the Auslander-Reiten quiver; and complexity and cohomology varieties, which has become a major theme in representation theory.

Author : Peter Webb
Publisher : Cambridge University Press
ISBN 13 : 1107162394
Total Pages : 339 pages
Book Rating : 4.96/5 ( download)

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Download or read book A Course in Finite Group Representation Theory written by Peter Webb and published by Cambridge University Press. This book was released on 2016-08-19 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. 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 representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.

Author : J. L. Alperin
Publisher : Cambridge University Press
ISBN 13 : 9780521449267
Total Pages : 198 pages
Book Rating : 4.6X/5 ( download)

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Download or read book Local Representation Theory written by J. L. Alperin and published by Cambridge University Press. This book was released on 1993-09-24 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this text is to present some of the key results in the representation theory of finite groups. In order to keep the account reasonably elementary, so that it can be used for graduate-level courses, Professor Alperin has concentrated on local representation theory, emphasising module theory throughout. In this way many deep results can be obtained rather quickly. After two introductory chapters, the basic results of Green are proved, which in turn lead in due course to Brauer's First Main Theorem. A proof of the module form of Brauer's Second Main Theorem is then presented, followed by a discussion of Feit's work connecting maps and the Green correspondence. The work concludes with a treatment, new in part, of the Brauer-Dade theory. As a text, this book contains ample material for a one semester course. Exercises are provided at the end of most sections; the results of some are used later in the text. Representation theory is applied in number theory, combinatorics and in many areas of algebra. This book will serve as an excellent introduction to those interested in the subject itself or its applications.

Author : Martin Burrow
Publisher : Academic Press
ISBN 13 : 1483258211
Total Pages : 196 pages
Book Rating : 4.18/5 ( download)

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Download or read book Representation Theory of Finite Groups written by Martin Burrow and published by Academic Press. This book was released on 2014-05-10 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. The succeeding chapters describe the features of representation theory of rings with identity and finite groups. These topics are followed by a discussion of some of the application of the theory of characters, along with some classical theorems. The last chapter deals with the construction of irreducible representations of groups. This book will be of great value to graduate students who wish to acquire some knowledge of representation theory.

Author : Jacques Thévenaz
Publisher : Oxford University Press
ISBN 13 : 9780198535874
Total Pages : 570 pages
Book Rating : 4.72/5 ( download)

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Download or read book G-algebras and Modular Representation Theory written by Jacques Thévenaz and published by Oxford University Press. This book was released on 1995 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a comprehensive treatment of the theory of G-Algebras and shows how it can be used to solve a number of problems about blocks, modules and almost split sequences. The new approach to modular representation theory of finite groups was developed mainly by Lluis Puig since the 1970s and has several characteristic features: unification of several theories (e.g. block theory and module theory) under a single concept, introduction of new invariants (e.g. source algebras and multiplicity modules) which shed new light on the whole, new point of view on some classical theorems (e.g. Brauer's second main theorem) yielding more precise results, deep structural results such as Puig's theory on nilpotent blocks.

Author : Steven H. Weintraub
Publisher : American Mathematical Soc.
ISBN 13 : 0821832220
Total Pages : 226 pages
Book Rating : 4.26/5 ( download)

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Download or read book Representation Theory of Finite Groups: Algebra and Arithmetic written by Steven H. Weintraub and published by American Mathematical Soc.. This book was released on 2003 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: ``We explore widely in the valley of ordinary representations, and we take the reader over the mountain pass leading to the valley of modular representations, to a point from which (s)he can survey this valley, but we do not attempt to widely explore it. We hope the reader will be sufficiently fascinated by the scenery to further explore both valleys on his/her own.'' --from the Preface Representation theory plays important roles in geometry, algebra, analysis, and mathematical physics. In particular, representation theory has been one of the great tools in the study and classification of finite groups. There are some beautiful results that come from representation theory: Frobenius's Theorem, Burnside's Theorem, Artin's Theorem, Brauer's Theorem--all of which are covered in this textbook. Some seem uninspiring at first, but prove to be quite useful. Others are clearly deep from the outset. And when a group (finite or otherwise) acts on something else (as a set of symmetries, for example), one ends up with a natural representation of the group. This book is an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra. The approach is to develop the requisite algebra in reasonable generality and then to specialize it to the case of group representations. Methods and results particular to group representations, such as characters and induced representations, are developed in depth. Arithmetic comes into play when considering the field of definition of a representation, especially for subfields of the complex numbers. The book has an extensive development of the semisimple case, where the characteristic of the field is zero or is prime to the order of the group, and builds the foundations of the modular case, where the characteristic of the field divides the order of the group. The book assumes only the material of a standard graduate course in algebra. It is suitable as a text for a year-long graduate course. The subject is of interest to students of algebra, number theory and algebraic geometry. The systematic treatment presented here makes the book also valuable as a reference.

Author : David A. Craven
Publisher : Springer Nature
ISBN 13 : 3030217922
Total Pages : 294 pages
Book Rating : 4.21/5 ( download)

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Download or read book Representation Theory of Finite Groups: a Guidebook written by David A. Craven and published by Springer Nature. This book was released on 2019-08-30 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an accessible introduction to the state of the art of representation theory of finite groups. Starting from a basic level that is summarized at the start, the book proceeds to cover topics of current research interest, including open problems and conjectures. The central themes of the book are block theory and module theory of group representations, which are comprehensively surveyed with a full bibliography. The individual chapters cover a range of topics within the subject, from blocks with cyclic defect groups to representations of symmetric groups. Assuming only modest background knowledge at the level of a first graduate course in algebra, this guidebook, intended for students taking first steps in the field, will also provide a reference for more experienced researchers. Although no proofs are included, end-of-chapter exercises make it suitable for student seminars.

Author : Pavel I. Etingof
Publisher : American Mathematical Soc.
ISBN 13 : 0821853511
Total Pages : 240 pages
Book Rating : 4.11/5 ( download)

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Download or read book Introduction to Representation Theory written by Pavel I. Etingof and published by American Mathematical Soc.. This book was released on 2011 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.

Author : James E. Humphreys
Publisher : Cambridge University Press
ISBN 13 : 9780521674546
Total Pages : 260 pages
Book Rating : 4.49/5 ( download)

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Download or read book Modular Representations of Finite Groups of Lie Type written by James E. Humphreys and published by Cambridge University Press. This book was released on 2006 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: A comprehensive treatment of the representation theory of finite groups of Lie type over a field of the defining prime characteristic.