Linear Representations of Groups

This textbook contains a comprehensive and detailed exposition of the fundamentals of the representation theory of groups, especially of finite groups and compact groups. The exposition is based on the decomposition of the two-sided regular representation. This enables the author to give not only an abstract description of the representations but also their realizations in function spaces, which is important for physical applications. As an example, the theory of Laplace spherical functions is treated. Some basic ideas of the representation theory of Lie groups are also given, as well as all the representations of the groups SU2 and SO3. The book contains numerous examples and exercises, some with solutions.

------ Reviews

This book is a short modern introduction to representation theory of groups. (...) Basic examples and exercises enable the reader to change over to explicit calculations.

• Zentralblatt MATH

This is a short and very readable introduction to finite-dimensional representation theory. A preliminary chapter gives the general flavor of the theory describing many examples and emphasizing the "harmonic analysis" aspects. (...) There are many exercises in the text; at the end of the book there is a list of some answers and solutions. (...) Even though the book is very short, its exposition is clear and unhurried. By concentrating on essential matters the author succeeds in getting across many of the fundamental ideas of the theory.

• Mathematical Reviews

Autorentext

Ernest B. Vinberg is a Professor of mathematics at the Moscow University.

Klappentext

This book gives an exposition of the fundamentals of the theory of linear representations of ?nite and compact groups, as well as elements of the t- ory of linear representations of Lie groups. As an application we derive the Laplace spherical functions. The book is based on lectures that I delivered in the framework of the experimental program at the Mathematics-Mechanics Faculty of Moscow State University and at the Faculty of Professional Skill Improvement. My aim has been to give as simple and detailed an account as possible of the problems considered. The book therefore makes no claim to completeness. Also, it can in no way give a representative picture of the modern state of the ?eld under study as does, for example, the monograph of A. A. Kirillov [3]. For a more complete acquaintance with the theory of representations of ?nite groups we recommend the book of C. W. Curtis and I. Reiner [2], and for the theory of representations of Lie groups, that of M. A. Naimark [6]. Introduction The theory of linear representations of groups is one of the most widely - pliedbranchesof algebra. Practically every timethatgroupsareencountered, their linear representations play an important role. In the theory of groups itself, linear representations are an irreplaceable source of examples and a tool for investigating groups. In the introduction we discuss some examples and en route we introduce a number of notions of representation theory. 0. Basic Notions 0. 1.

Inhalt

Preface Introduction 0. Basic Notions I. General Properties of Representations 1. Invariant Subspaces 2. Complete Reducibility of Representations of Compact Groups 3. Basic Operations on Representations 4. Properties of Irreducible Complex Representations II. Representations of Finite Groups 5. Decomposition of the Regular Representation 6. Orthogonality Relations III. Representations of Compact Groups 7. The Groups SU2 and SO3 8. Matrix Elements of Compact Groups9. The Laplace Spherical Functions IV. Representations of Lie Groups10. General Properties of Homomorphisms and Representations of Lie Groups 11. Representations of SU2 and SO3Appendices A1 Presentation of Groups By Means ofGenerators and Relations A2 Tensor Products A3 The Convex Hull of a Compact Set A4 Conjugate Elements in Groups Answers and Hints to Exercises List of Notations References Index