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G Algebras And Modular Representation Theory
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Book Synopsis G-algebras and Modular Representation Theory by : Jacques Thévenaz
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.
Book Synopsis Modular Representation Theory by : D. Benson
Download or read book Modular Representation Theory written by D. Benson and published by Springer. This book was released on 2008-07-22 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.
Book Synopsis Modular Representation Theory of Finite Groups by : Peter Schneider
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.
Book Synopsis Local Representation Theory by : J. L. Alperin
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.
Book Synopsis Group Representations by : Gregory Karpilovsky
Download or read book Group Representations written by Gregory Karpilovsky and published by North Holland. This book was released on 1992 with total page 980 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is divided into three parts. Part I provides the foundations of the theory of modular representations. Special attention is drawn to the Brauer-Swan theory and the theory of Brauer characters. A detailed investigation of quadratic, symplectic and symmetric modules is also provided. Part II is devoted entirely to the Green theory: vertices and sources, the Green correspondence, the Green ring, etc. In Part III, permutation modules are investigated with an emphasis on the study of p-permutation modules and Burnside rings. The material is developed with sufficient attention to detail so that it can easily be read by the novice, although its chief appeal will be to specialists. A number of the results presented in this volume have almost certainly never been published before.
Book Synopsis Modular Representation Theory of Finite and p-Adic Groups by : Wee Teck Gan
Download or read book Modular Representation Theory of Finite and p-Adic Groups written by Wee Teck Gan and published by World Scientific. This book was released on 2015-02-13 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is an outgrowth of the program Modular Representation Theory of Finite and p-Adic Groups held at the Institute for Mathematical Sciences at National University of Singapore during the period of 1–26 April 2013. It contains research works in the areas of modular representation theory of p-adic groups and finite groups and their related algebras. The aim of this volume is to provide a bridge — where interactions are rare between researchers from these two areas — by highlighting the latest developments, suggesting potential new research problems, and promoting new collaborations. It is perhaps one of the few volumes, if not only, which treats such a juxtaposition of diverse topics, emphasizing their common core at the heart of Lie theory. Contents:Modular Representations of Finite Reductive Groups (Marc Cabanes)ℓ-Modular Representations of p-Adic Groups (ℓ ≠ p) (Vincent Sécherre)p-Modular Representations of p-Adic Groups (Florian Herzig)Representation Theory and Cohomology of Khovanov–Lauda–Rouquier Algebras (Alexander S Kleshchev)Cyclotomic Quiver Hecke Algebras of Type A (Andrew Mathas) Readership: Graduate students and professional mathematicians interested in modular representation theory. Key Features:Contains a survey of modular representation theory of finite groups of Lie type, with a description of recent progress and outstanding conjecturesCovers the modular representation theory of p-adic groups in both defining and non-defining characteristic which is being pursued in the modular Langlands programIntroduces the increasingly popular representation theory of Khovanov–Lauda–Rouquier algebras and the graded representation theory of cyclotomic Hecke algebrasSuitable for graduate students as well as mathematical researchers who desire to learn about representation theory in these areasKeywords:Modular Representation Theory;Reductive Groups;Modular Langlands Program;Khovanov–Lauda–Rouquier Algebras;Cyclotomic Hecke Algebras
Book Synopsis Modular Representation Theory of Finite Groups by : Michael J. Collins
Download or read book Modular Representation Theory of Finite Groups written by Michael J. Collins and published by Walter de Gruyter. This book was released on 2011-07-11 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 1998. 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 brief was to present material at the leading edge of research but accessible to graduate students working in the field. The first three articles are substantial expansions of their lectures, and each provides a complete account of a significant area of the subject together with an extensive bibliography. The remaining articles are based on some of the other lectures given at the symposium; some again are full surveys of the topic covered while others are short, but complete, research articles. The opportunity has been taken to produce a book of enduring value so that this is not a conference proceedings in the conventional sense. Material has been updated so that this book, through its own content and in its extensive bibliographies, will serve as an invaluable resource for all those working in the area, whether established researchers or graduate students who wish to gain a general knowledge of the subject starting from a single source.
Book Synopsis Group Representation Theory: Modular representation theory by : Larry L. Dornhoff
Download or read book Group Representation Theory: Modular representation theory written by Larry L. Dornhoff and published by . This book was released on 1971 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Course in Finite Group Representation Theory by : Peter Webb
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.
Book Synopsis Representation Theory of Finite Groups by : Martin Burrow
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.
Book Synopsis Modular Representation Theory by : D. Benson
Download or read book Modular Representation Theory written by D. Benson and published by . This book was released on 2014-01-15 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Modular Representations of Finite Groups by :
Download or read book Modular Representations of Finite Groups written by and published by Academic Press. This book was released on 1977-03-14 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modular Representations of Finite Groups
Book Synopsis Representations of Finite Groups by : Hirosi Nagao
Download or read book Representations of Finite Groups written by Hirosi Nagao and published by Elsevier. This book was released on 2014-05-10 with total page 443 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 modules. This text then provides the theory of algebras, including theories of simple algebras, Frobenius algebras, crossed products, and Schur indices with representation-theoretic versions of them. Other chapters include a survey of the fundamental theory of modular representations, with emphasis on Brauer characters. This book discusses as well the module-theoretic representation theory due to Green and includes some topics such as Burry–Carlson's theorem and Scott modules. The final chapter deals with the fundamental results of Brauer on blocks and Fong's theory of covering, and includes some approaches to them. This book is a valuable resource for readers who are interested in the various approaches to the study of the representations of groups.
Book Synopsis Modular Lie Algebras and their Representations by : H. Strade
Download or read book Modular Lie Algebras and their Representations written by H. Strade and published by CRC Press. This book was released on 2020-08-11 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents an introduction to the structure and representation theory of modular Lie algebras over fields of positive characteristic. It introduces the beginner to the theory of modular Lie algebras and is meant to be a reference text for researchers.
Book Synopsis Representation Theories and Algebraic Geometry by : A. Broer
Download or read book Representation Theories and Algebraic Geometry written by A. Broer and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 455 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 12 lectures presented in Representation Theories and Algebraic Geometry focus on the very rich and powerful interplay between algebraic geometry and the representation theories of various modern mathematical structures, such as reductive groups, quantum groups, Hecke algebras, restricted Lie algebras, and their companions. This interplay has been extensively exploited during recent years, resulting in great progress in these representation theories. Conversely, a great stimulus has been given to the development of such geometric theories as D-modules, perverse sheafs and equivariant intersection cohomology. The range of topics covered is wide, from equivariant Chow groups, decomposition classes and Schubert varieties, multiplicity free actions, convolution algebras, standard monomial theory, and canonical bases, to annihilators of quantum Verma modules, modular representation theory of Lie algebras and combinatorics of representation categories of Harish-Chandra modules.
Book Synopsis Representation Theory of Finite Groups: Algebra and Arithmetic by : Steven H. Weintraub
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.
Book Synopsis Modular Representations of Finite Groups of Lie Type by : James E. Humphreys
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.
