| 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 |
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