An Introduction to the Representation Theory of Groups

[BOOK DESCRIPTION]

Representation theory is an important part of modern mathematics, not only as a subject in its own right but also as a tool for many applications. It provides a means for exploiting symmetry, making it particularly useful in number theory, algebraic geometry, and differential geometry, as well as classical and modern physics. The goal of this book is to present, in a motivated manner, the basic formalism of representation theory as well as some important applications. The style is intended to allow the reader to gain access to the insights and ideas of representation theory--not only to verify that a certain result is true, but also to explain why it is important and why the proof is natural. The presentation emphasizes the fact that the ideas of representation theory appear, sometimes in slightly different ways, in many contexts. Thus the book discusses in some detail the fundamental notions of representation theory for arbitrary groups. It then considers the special case of complex representations of finite groups and discusses the representations of compact groups, in both cases with some important applications.There is a short introduction to algebraic groups as well as an introduction to unitary representations of some noncompact groups. The text includes many exercises and examples.

[TABLE OF CONTENTS]

Chapter 1 Introduction and motivation              1  (12)
  1.1 Presentation                                 3  (1)
  1.2 Four motivating statements                   4  (4)
  1.3 Prerequisites and notation                   8  (5)
Chapter 2 The language of representation theory    13 (114)
  2.1 Basic language                               13 (8)
  2.2 Formalism: changing the space                21 (21)
  2.3 Formalism: changing the group                42 (23)
  2.4 Formalism: changing the field                65 (3)
  2.5 Matrix representations                       68 (2)
  2.6 Examples                                     70 (10)
  2.7 Some general results                         80 (41)
  2.8 Some Clifford theory                         121(3)
  2.9 Conclusion                                   124(3)
Chapter 3 Variants                                 127(32)
  3.1 Representations of algebras                  127(5)
  3.2 Representations of Lie algebras              132(7)
  3.3 Topological groups                           139(6)
  3.4 Unitary representations                      145(14)
Chapter 4 Linear representations of finite         159(110)
groups
  4.1 Maschke's Theorem                            159(4)
  4.2 Applications of Maschke's Theorem            163(6)
  4.3 Decomposition of representations             169(21)
  4.4 Harmonic analysis on finite groups           190(10)
  4.5 Finite abelian groups                        200(8)
  4.6 The character table                          208(32)
  4.7 Applications                                 240(22)
  4.8 Further topics                               262(7)
Chapter 5 Abstract representation theory of        269(50)
compact groups
  5.1 An example: the circle group                 269(3)
  5.2 The Haar measure and the regular             272(16)
  representation of a locally compact group
  5.3 The analogue of the group algebra            288(6)
  5.4 The Peter邑eyl Theorem                       294(10)
  5.5 Characters and matrix coefficients for       304(6)
  compact groups
  5.6 Some first examples                          310(9)
Chapter 6 Applications of representations of       319(36)
compact groups
  6.1 Compact Lie groups are matrix groups         319(5)
  6.2 The Frobenius-Schur indicator                324(8)
  6.3 The Larsen alternative                       332(12)
  6.4 The hydrogen atom                            344(11)
Chapter 7 Other groups: a few examples             355(54)
  7.1 Algebraic groups                             355(14)
  7.2 Locally compact groups: general remarks      369(2)
  7.3 Locally compact abelian groups               371(5)
  7.4 A non-abelian example: SL2(R)                376(33)
Appendix A Some useful facts                       409(12)
  A.1 Algebraic integers                           409(5)
  A.2 The spectral theorem                         414(6)
  A.3 The Stone邑eierstrass Theorem                420(1)
Bibliography                                       421(4)
Index                                              425