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Representations of Finite and Compact Groups
Barry Simon, California Institute of Technology, Pasadena, CA
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1996; 266 pp; hardcover
Volume: 10
ISBN-10: 0-8218-0453-7
ISBN-13: 978-0-8218-0453-7
List Price: US$45 Member Price: US$36
Order Code: GSM/10

Barry Simon, I.B.M. Professor of Mathematics and Theoretical Physics at the California Institute of Technology, is the author of several books, including such classics as Methods of Mathematical Physics (with M. Reed) and Functional Integration and Quantum Physics. This new book, based on courses given at Princeton, Caltech, ETH-Zurich, and other universities, is an introductory textbook on representation theory. According to the author, "Two facets distinguish my approach. First, this book is relatively elementary, and second, while the bulk of the books on the subject is written from the point of view of an algebraist or a geometer, this book is written with an analytical flavor".

The exposition in the book centers around the study of representation of certain concrete classes of groups, including permutation groups and compact semisimple Lie groups. It culminates in the complete proof of the Weyl character formula for representations of compact Lie groups and the Frobenius formula for characters of permutation groups. Extremely well tailored both for a one-year course in representation theory and for independent study, this book is an excellent introduction to the subject which, according to the author, is unique in having "so much innate beauty so close to the surface".

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Reviews

"Contains a very good explanation of representation theory of finite and compact groups and can be recommended to everyone for learning or teaching representation theory."

-- Zentralblatt MATH

"This is indeed a nice book ... I would recommend it precisely for the graduate course [that] I am teaching now, "Representation Theory" ... I very much like the hands-on approach and the very explicit formulae that are given ... Professor Simon has done an excellent job on this beautiful material."

-- Tudor Ratiu, University of California, Santa Cruz

"Can be recommended as a base for courses about representations of finite groups and finite-dimensional representations of Lie groups."

-- Mathematical Reviews

"This book closes a gap that impeded instruction in the principles of the theory of group representations in the past ... it would also tend to close the gap between the interested student of the subject and the advanced researcher keen on bare facts ... a most readable account of a subject of enduring fascination."

-- Monatshefte für Mathematik

• Preface
• Chapters VIII, VII, and III
• Fundamentals of group represetnations (Chapter 2)
• Introduction
• Groups and counting principles (Chapter I)
• Fundamentals of group representations (Chapter II)
• Abstract theory of representations of finite groups (Chapter III)
• Representations of concrete finite groups. I: Abelian and Clifford groups (Chapter IV)
• Representations of concrete finite groups. II: Semidirect products and induced representations (Chapter V)
• Concrete representations of finite groups. III: The symmetric groups (Chapter VI)
• Compact groups (Chapter VII)
• The structure of compact semisimple groups (Chapter VIII)
• The representations of compact semisimple groups (Chapter IX)
• Multilinear algebra (Appendix A)
• The analysis of self-adjoint Hilbert-Schmidt operators (Appendix B)
• Bibliography
• Index