目錄
Preface
Part Ⅰ Compact Groups
1 Haar Measure
2 Schur Orthogonality
3 Compact Operators
4 The Peter-Weyl Theorem
Part Ⅱ Compact Lie Groups
5 Lie Subgroups of GL(n, C)
6 Vector Fields
7 Left-Invariant Vector Fields
8 The Exponential Map
9 Tensors and Universal Properties
10 The Universal Enveloping Algebra
11 Extension of Scalars
12 Representations of sl(2, C)
13 The Universal Cover
14 The Local Frobenius Theorem
15 Tori
16 Geodesics and Maximal Tori
17 The Weyl Integration Formula
18 The Root System
19 Examples of Root Systems
20 Abstract Weyl Groups
21 Highest Weight Vectors
22 The Weyl Character Formula
23 The Fundamental Group
Part Ⅲ Noncompact Lie Groups
24 Complexiflcation
25 Coxeter Groups
26 The Borel Subgroup
27 The Bruhat Decomposition
28 Symmetric Spaces
29 Relative Root Systems
30 Embeddings of Lie Groups
31 Spin
Part Ⅳ Duality and Other Topics
32 Mackey Theory
33 Characters of GL(n, C)
34 Duality Between Sk and GL(n, C)
35 The Jacobi-Trudi Identity
36 Schur Polynomials and GL(n, C)
37 Schur Polynomials and Sk
38 The Cauchy Identity
39 Random Matrix Theory
40 Symmetric Group Branching Rules and Tableaux
41 Unitary Branching Rules and Tableaux
42 Minors of Toeplitz Matrices
43 The Involution Model for Sk
44 Some Symmetric Algebras
45 Gelfand Pairs
46 Hecke Algebras
47 The Philosophy of Cusp Forms
48 Cohomology of Grassmannians
Appendix: Sage
References
Index
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