Projective Representations Of Finite Groups

Author : Gregory Karpilovsky
Publisher : Unknown
Page : 672 pages
File Size : 41,7 Mb
Release : 1985
Category : Mathematics
ISBN : UOM:39015014357761

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Projective Representations of Finite Groups by Gregory Karpilovsky Pdf

This book presents a systematic account of this topic, from the classical foundations established by Schur 80 years ago to current advances and developments in the field. This work focuses on general methods and builds theory solidly on the study of modules over twisted group algebras, and provides a wide range of skill-sharpening mathematical techniques applicable to this subject. Offers an understanding of projective representations of finite groups for algebraists, number theorists, mathematical researchers studying modern algebra, and theoretical physicists.

Author : Gregory Karpilovsky
Publisher : Unknown
Page : 664 pages
File Size : 49,7 Mb
Release : 2024-07-03
Category : Electronic
ISBN : 0598040110

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Projective Representations of Finite Groups by Gregory Karpilovsky Pdf

Author : Hirosi Nagao,Yukio Tsushima
Publisher : Elsevier
Page : 443 pages
File Size : 54,5 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483269931

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Representations of Finite Groups by Hirosi Nagao,Yukio Tsushima Pdf

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.

Author : J. S. Lomont
Publisher : Academic Press
Page : 359 pages
File Size : 54,6 Mb
Release : 2014-05-12
Category : Mathematics
ISBN : 9781483268965

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Applications of Finite Groups by J. S. Lomont Pdf

Applications of Finite Groups focuses on the applications of finite groups to problems of physics, including representation theory, crystals, wave equations, and nuclear and molecular structures. The book first elaborates on matrices, groups, and representations. Topics include abstract properties, applications, matrix groups, key theorem of representation theory, properties of character tables, simply reducible groups, tensors and invariants, and representations generated by functions. The text then examines applications and subgroups and representations, as well as subduced and induced representations, fermion annihilation and creation operators, crystallographic point groups, proportionality tensors in crystals, and nonrelativistic wave equations. The publication takes a look at space group representations and energy bands, symmetric groups, and applications. Topics include molecular and nuclear structures, multiplet splitting in crystalline electric fields, construction of irreducible representations of the symmetric groups, and reality of representations. The manuscript is a dependable source of data for physicists and researchers interested in the applications of finite groups.

Author : Peter Webb
Publisher : Cambridge University Press
Page : 339 pages
File Size : 40,5 Mb
Release : 2016-08-19
Category : Mathematics
ISBN : 9781107162396

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A Course in Finite Group Representation Theory by Peter Webb Pdf

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 : Tullio Ceccherini-Silberstein,Fabio Scarabotti,Filippo Tolli
Publisher : Springer Nature
Page : 347 pages
File Size : 47,7 Mb
Release : 2022-11-29
Category : Mathematics
ISBN : 9783031138737

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Representation Theory of Finite Group Extensions by Tullio Ceccherini-Silberstein,Fabio Scarabotti,Filippo Tolli Pdf

This monograph adopts an operational and functional analytic approach to the following problem: given a short exact sequence (group extension) 1 → N → G → H → 1 of finite groups, describe the irreducible representations of G by means of the structure of the group extension. This problem has attracted many mathematicians, including I. Schur, A.H. Clifford, and G. Mackey and, more recently, M. Isaacs, B. Huppert, Y.G. Berkovich & E.M. Zhmud, and J.M.G. Fell & R.S. Doran. The main topics are, on the one hand, Clifford Theory and the Little Group Method (of Mackey and Wigner) for induced representations, and, on the other hand, Kirillov’s Orbit Method (for step-2 nilpotent groups of odd order) which establishes a natural and powerful correspondence between Lie rings and nilpotent groups. As an application, a detailed description is given of the representation theory of the alternating groups, of metacyclic, quaternionic, dihedral groups, and of the (finite) Heisenberg group. The Little Group Method may be applied if and only if a suitable unitary 2-cocycle (the Mackey obstruction) is trivial. To overcome this obstacle, (unitary) projective representations are introduced and corresponding Mackey and Clifford theories are developed. The commutant of an induced representation and the relative Hecke algebra is also examined. Finally, there is a comprehensive exposition of the theory of projective representations for finite Abelian groups which is applied to obtain a complete description of the irreducible representations of finite metabelian groups of odd order.

Author : Peter Schneider
Publisher : Springer Science & Business Media
Page : 183 pages
File Size : 48,6 Mb
Release : 2012-11-27
Category : Mathematics
ISBN : 9781447148326

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Modular Representation Theory of Finite Groups by Peter Schneider Pdf

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 : James E. Humphreys
Publisher : Cambridge University Press
Page : 260 pages
File Size : 42,8 Mb
Release : 2006
Category : Mathematics
ISBN : 0521674549

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Modular Representations of Finite Groups of Lie Type by James E. Humphreys Pdf

A comprehensive treatment of the representation theory of finite groups of Lie type over a field of the defining prime characteristic.

Author : Jean-Pierre Serre
Publisher : Springer Science & Business Media
Page : 178 pages
File Size : 40,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468494587

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Linear Representations of Finite Groups by Jean-Pierre Serre Pdf

This book consists of three parts, rather different in level and purpose. The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and characters. The second part is a course given in 1966 to second-year students of l’Ecole Normale. It completes in a certain sense the first part. The third part is an introduction to Brauer Theory.

Author : Martin Burrow
Publisher : Academic Press
Page : 196 pages
File Size : 51,9 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483258218

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Representation Theory of Finite Groups by Martin Burrow Pdf

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 : Jean Pierre Serre
Publisher : Unknown
Page : 170 pages
File Size : 41,7 Mb
Release : 1996
Category : Electronic
ISBN : OCLC:806314847

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Linear Representations of Finite Groups by Jean Pierre Serre Pdf

Author : Aleksandr Sergeevich Kleshchëv,Vladimir Shchigolev
Publisher : American Mathematical Soc.
Page : 123 pages
File Size : 42,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821874318

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Modular Branching Rules for Projective Representations of Symmetric Groups and Lowering Operators for the Supergroup $Q(n)$ by Aleksandr Sergeevich Kleshchëv,Vladimir Shchigolev Pdf

There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation theory of the symmetric group and representation theory of the algebraic supergroup $Q(n)$ via appropriate Schur (super)algebras and Schur functors. The second approach follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras and connects projective representation theory of symmetric groups in characteristic $p$ to the crystal graph of the basic module of the twisted affine Kac-Moody algebra of type $A_{p-1}^{(2)}$. The goal of this work is to connect the two approaches mentioned above and to obtain new branching results for projective representations of symmetric groups.

Author : I. Martin Isaacs
Publisher : American Mathematical Soc.
Page : 322 pages
File Size : 40,6 Mb
Release : 2006-11-21
Category : Mathematics
ISBN : 9780821842294

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Character Theory of Finite Groups by I. Martin Isaacs Pdf

Character theory is a powerful tool for understanding finite groups. In particular, the theory has been a key ingredient in the classification of finite simple groups. Characters are also of interest in their own right, and their properties are closely related to properties of the structure of the underlying group. The book begins by developing the module theory of complex group algebras. After the module-theoretic foundations are laid in the first chapter, the focus is primarily on characters. This enhances the accessibility of the material for students, which was a major consideration in the writing. Also with students in mind, a large number of problems are included, many of them quite challenging. In addition to the development of the basic theory (using a cleaner notation than previously), a number of more specialized topics are covered with accessible presentations. These include projective representations, the basics of the Schur index, irreducible character degrees and group structure, complex linear groups, exceptional characters, and a fairly extensive introduction to blocks and Brauer characters. This is a corrected reprint of the original 1976 version, later reprinted by Dover. Since 1976 it has become the standard reference for character theory, appearing in the bibliography of almost every research paper in the subject. It is largely self-contained, requiring of the reader only the most basic facts of linear algebra, group theory, Galois theory and ring and module theory.

Author : Caroline Gruson,Vera Serganova
Publisher : Springer
Page : 223 pages
File Size : 46,8 Mb
Release : 2018-10-23
Category : Mathematics
ISBN : 9783319982717

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A Journey Through Representation Theory by Caroline Gruson,Vera Serganova Pdf

This text covers a variety of topics in representation theory and is intended for graduate students and more advanced researchers who are interested in the field. The book begins with classical representation theory of finite groups over complex numbers and ends with results on representation theory of quivers. The text includes in particular infinite-dimensional unitary representations for abelian groups, Heisenberg groups and SL(2), and representation theory of finite-dimensional algebras. The last chapter is devoted to some applications of quivers, including Harish-Chandra modules for SL(2). Ample examples are provided and some are revisited with a different approach when new methods are introduced, leading to deeper results. Exercises are spread throughout each chapter. Prerequisites include an advanced course in linear algebra that covers Jordan normal forms and tensor products as well as basic results on groups and rings.

Author : Peter Norman Hoffman,J. F. Humphreys
Publisher : Oxford University Press
Page : 322 pages
File Size : 41,8 Mb
Release : 1992
Category : Mathematics
ISBN : 0198535562

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Projective Representations of the Symmetric Groups by Peter Norman Hoffman,J. F. Humphreys Pdf

The study of the symmetric groups forms one of the basic building blocks of modern group theory. This book presents information currently known on the projective representations of the symmetric and alternating groups. Special emphasis is placed on the theory of Q-functions and skew Q-functions.