Author : Ilʹi︠a︡ Iosifovich Pi︠a︡tet︠s︡kiĭ-Shapiro
Publisher : American Mathematical Soc.
Page : 84 pages
File Size : 43,59 MB
Release : 1983
Category : Mathematics
ISBN : 0821850199
These are lecture notes of a course given at Tel-Aviv University. The aim of these notes is to present the theory of representations of GL(2, K) where K is a finite field. However, the presentation of the material has in mind the theory of infinite dimensional representations of GL(2, K) for local fields K.
Author : Il_i_a Iosifovich Pi_atet_ski_-Shapiro
Publisher : American Mathematical Soc.
Page : 86 pages
File Size : 47,72 MB
Release : 1983-12-31
Category : Mathematics
ISBN : 9780821853757
Author : Zeph Grunschlag
Publisher :
Page : 0 pages
File Size : 48,65 MB
Release : 1992
Category :
ISBN :
Author : Pavel I. Etingof
Publisher : American Mathematical Soc.
Page : 240 pages
File Size : 35,73 MB
Release : 2011
Category : Mathematics
ISBN : 0821853511
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 : Peter Webb
Publisher : Cambridge University Press
Page : 339 pages
File Size : 39,39 MB
Release : 2016-08-19
Category : Mathematics
ISBN : 1107162394
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 : H. Jacquet
Publisher : Springer
Page : 156 pages
File Size : 39,84 MB
Release : 2006-11-15
Category : Mathematics
ISBN : 3540376127
Author : Gurjyot Kaur
Publisher :
Page : 0 pages
File Size : 40,34 MB
Release : 2020
Category :
ISBN :
Let $G=\GL(2, \mathbb{F}_q)$ denote the group of invertible $2\times 2$ matrices over the finite field $\mathbb{F}_q$, where $q$ is the power of a prime number. This thesis investigates the complex irreducible representations of $G$. Chapter 2 gives an overview of the general theory of finite-dimensional representations of a finite group over the complex numbers. In Chapter 3, we discuss the different types of conjugacy classes in $G$. It will turn out that there are four types, which altogether give $q^2-1$ conjugacy classes. By general theory, the number of irreducible representations distinguished up to isomorphism is also equal to $q^2-1$. These representations are explicitly constructed in Chapter 4. Some of them are derived from the permutation representation of $G$ on the projective line $\mathbb{P}^1(\FF_q)$, while the rest are induced from smaller subgroups of $G$. In the concluding chapter, we compute the irreducible decompositions of the tensor products of some of these irreducible representations by using the character table for irreducible $G$-representations.
Author : Benjamin Steinberg
Publisher : Springer Science & Business Media
Page : 166 pages
File Size : 35,78 MB
Release : 2011-10-23
Category : Mathematics
ISBN : 1461407761
This book is intended to present group representation theory at a level accessible to mature undergraduate students and beginning graduate students. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. This book is intended for a 3rd/4th undergraduate course or an introductory graduate course on group representation theory. However, it can also be used as a reference for workers in all areas of mathematics and statistics.
Author : J. L. Alperin
Publisher : Cambridge University Press
Page : 198 pages
File Size : 46,72 MB
Release : 1993-09-24
Category : Mathematics
ISBN : 9780521449267
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 : Cédric Bonnafé
Publisher : Springer Science & Business Media
Page : 196 pages
File Size : 25,57 MB
Release : 2010-10-08
Category : Mathematics
ISBN : 0857291572
Deligne-Lusztig theory aims to study representations of finite reductive groups by means of geometric methods, and particularly l-adic cohomology. Many excellent texts present, with different goals and perspectives, this theory in the general setting. This book focuses on the smallest non-trivial example, namely the group SL2(Fq), which not only provides the simplicity required for a complete description of the theory, but also the richness needed for illustrating the most delicate aspects. The development of Deligne-Lusztig theory was inspired by Drinfeld's example in 1974, and Representations of SL2(Fq) is based upon this example, and extends it to modular representation theory. To this end, the author makes use of fundamental results of l-adic cohomology. In order to efficiently use this machinery, a precise study of the geometric properties of the action of SL2(Fq) on the Drinfeld curve is conducted, with particular attention to the construction of quotients by various finite groups. At the end of the text, a succinct overview (without proof) of Deligne-Lusztig theory is given, as well as links to examples demonstrated in the text. With the provision of both a gentle introduction and several recent materials (for instance, Rouquier's theorem on derived equivalences of geometric nature), this book will be of use to graduate and postgraduate students, as well as researchers and lecturers with an interest in Deligne-Lusztig theory.
