Linear Representations of Groups: Translated from the Russian by A. Iacob

ISBN-13: 9783034800624 / Angielski / Miękka / 2010 / 146 str.

Ernest B. Vinberg

Opis

This book presents 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.

Recenzje

From the reviews:

"Vinberg's book should be seen as preliminary to far more ambitious and weighty texts ... . It takes the reader from raw basics, done very nicely, through the obligatory discussion of the regular representation ... to some excellent stuff on SU₂ and SO₃, with a discussion of Laplace's spherical functions thrown in. The presence of a huge number of guiding examples is a major asset ... as is the large collection of exercises for the reader. It is indeed a marvelous introduction to representation theory!" (Michael Berg, The Mathematical Association of America, February, 2011)

Spis treści

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

O autorze

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

Blurb

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