有限羣的表示論

《有限羣的表示論》是一部為大學高年級本科生和低年級研究生編寫的教科書，內容主要涉及羣表示論的各個方面。閲讀本書所需背景知識包括線性代數，羣論、環論基礎知識，書中有意省略模理論，Wedderburn理論和張量積等內容，取而代之的加入離散傅里葉分析。 ^[1] 

1 Introduction

2 Review of Linear Algebra

2.1 Basic Definitions and Notation

2.2 Complex Inner Product Spaces

2.3 Further Notions from Linear Algebra

3 Group Rep resentations

3.1 Basic Definitions and First Examples

3.2 Maschke's Theorem and Complete Reducibility

4 Character Theory and the Orthogonality Relations

4.1 Morphisms of Representations

4.2 The Orthogonality Relations

4.3 Characters and Class Functions

4.4 The Regular Representation

4.5 Representations of Abelian Groups

5 Fourier Analysis on Finite Groups

5.1 Periodic Functions on Cyclic Groups

5.2 The Convolution Product

5.3 Fourier Analysis on Finite Abelian Groups

5.4 An Application to Graph Theory

5.5 Fourier Analysis on Non—abelian Groups

6 Bumside's Theorem

6.1 A Little Number Theory

6.2 The Dimension Theorem

6.3 Burnside's Theorem

7 Group Actions and Permutation Representations

7.1 Group Actions

7.2 Permutation Representations

7.3 The Centralizer Algebra and Gelfand Pairs

8 Induced Representations

8.1 Induced Characters and Frobenius Reciprocity

8.2 Induced Representations

8.3 Mackey's Irreducibility Criterion

9 Another Theorem of Burnside

9.1 Conjugate Represen tations

10 Representation Theory of the Symmetric Group

10.1 Partitions and Tableaux

10.2 Constructing thelneducible Representations

11 Probability and Random Walks on Groups

11.1 Probabilities on Groups

11.2 Random Walks on Finite Groups

11.3 CardShuffling

11.3.1 The Riffle Shuffle

11.4 The Spectrum and the Upper Bound Lemma

References

Index ^[1]