Part Ⅰ Compact Groups 3
1 Haar Measure 3
2 Schur Orthogonality 7
3 Compact Operators 19
4 The Peter-Weyl Theorem 23
Part Ⅱ Compact Lie Groups 31
5 Lie Subgroups of GL（n,C） 31
6 Vector Fields 39
7 Left-Invariant Vector Fields 45
8 The Exponential Map 51
9 Tensors and Universal Properties 57
10 The Universal Enveloping Algebra 61
11 Extension of Scalars 67
12 Representations of sl（2,C） 71
13 The Universal Cover 81
14 The Local Frobenius Theorem 93
15 Tori 101
16 Geodesics and Maximal Tori 109
17 The Weyl Integration Formula 123
18 The Root System 129
19 Examples of Root Systems 145
20 Abstract Weyl Groups 157
21 Highest Weight Vectors 169
22 The Weyl Character Formula 177
23 The Fundamental Group 191
Part Ⅲ Noncompact Lie Groups 205
24 Complexification 205
25 Coxeter Groups 213
26 The Borel Subgroup 227
27 The Bruhat Decomposition 243
28 Symmetric Spaces 257
29 Relative Root Systems 281
30 Embeddings of Lie Groups 303
31 Spin 319
Part Ⅳ Duality and Other Topics 337
32 Mackey Theory 337
33 Characters of GL（n,C） 349
34 Duality Between Sk and GL（n,C） 355
35 The Jacobi-Trudi Identity 365
36 Schur Polynomials and GL（n,C） 379
37 Schur Polynomials and Sk 387
38 The Cauchy Identity 395
39 Random Matrix Theory 407
40 Symmetric Group Branching Rules and Tableaux 419
41 Unitary Branching Rules and Tableaux 427
42 Minors of Toeplitz Matrices 437
43 The Involution Model for Sk 445
44 Some Symmetric Algebras 455
45 Gelfand Pairs 461
46 Hecke Algebras 471
47 The Philosophy of Cusp Forms 485
48 Cohomology of Grassmannians 517
Appendix:Sage 529
References 535
Index 545