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 Complexification

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