Part Ⅰ
Representations and Characters
1 Generalities on linear representations
1.1 Definitions
1.2 Basic examples
1.3 Subrepresentations
1.4 Irreducible representations
1.5 Tensor product of two representations
1.6 Symmetiic square and alternating square
2 Character theory
2.1 The character of a representation
2.2 Schur’’s lemma; basic applications
2.3 Orthogonality’’reiations for characters
2.4 Decomposition of the regular representation
2.5 Number of irreducible representations
2.6 Canonical decomposition of a representation
2.7 Explicit decomposition of a representation
3 Subgroups, products, induced representations
3.1 Abelian subgroups
3.2 Product of two groups
3.3 Induced representations
4 Compact groups
4.1 Compact groups
4.2 lnvariant measure on a compact group
4.3 Linear representations of compact groups
5 Examples
5.1 The cyclic Group Cn
5.2 The group C
5.3 The dihedral group D
5.4 The group Dn
5.5 The group D
5.6 The group D
5.7 The alternating group
5.8 The symmetric group
5.9 The group of the cube
Bibliography: Part I
Part Ⅱ Representations in Characteristic Zero
6 The group algebra
6,1 Representations and modules
6.2 Decomposition of C[G]
6.3 The center of C[G]
6.4 Basic properties of integers
6.5 lntegrality properties of characters. Applications
7 Induced representations; Mackey’’s criterion
7.1 Induction
7.2 The character of an induced representation; the reciprocity formula
7.3 Restriction to subgroups
7.4 Mackey’’s irreducibility criterion
8 Examples of induced representations
8.1 Normal subgroups; applications to the degrees of the irreducible representations
8.2 Semidirect products by an abelian group
8.3 A review of some classes of finite groups
8.4 Sylow’’s theorem
8.5 Linear representations of supersolvable groups
9 Artin’’s theorem
9.1 The ring R(G)
9,2 Statement of Artin’’s theorem
9.3 First proof
9.4 Second proof of (i) (ii)
10 A theorem of Brauer
10.1 p-regular elements;p-elementary subgroups
10.2 Induced characters arising from p-elementary subgroups
10.3 Construction of characters
10.4 Proof of theorems 18 and 18’’
10,5 Brauer’’s theorem
……
part Ⅲ Introduction to Brauer Theory
Appendix
Bibliography:Part Ⅲ
Index of notation
Index of terminology
