viii CONTENTS CHAPTER IV. Representations of Concrete Finite Groups. I: Abelian and Clifford Groups 65 1 The structure of finite abelian groups 65 2 Representations of abelian groups 67 3 The Clifford group 68 CHAPTER V. Representations of Concrete Finite Groups. II: Semidirect Products and Induced Representations 77 1 Frobenius theory of semidirect products 77 2 Examples of the semidirect product theory 81 3 Induced representations 83 4 The Frobenius character formula 85 5 The Frobenius reciprocity theorem 89 6 Mackey irreducibility criterion 91 7 Semidirect products, revisited 93 CHAPTER VI. Representations of Concrete Finite Groups. Ill: The Symmetric Groups 95 1 Permutations and classes 95 2 Young frames and Young tableaux 96 3 Projections in A(Sn): Classification of representations 101 4 Branching relations 108 5 The Frobenius character formula 109 6 Consequences of the character formula 117 CHAPTER VII. Compact Groups 121 1 C°°-manifolds: A review 121 2 Lie groups and Lie algebras 128 3 Haar measure on Lie groups 133 4 Matrix groups 135 5 The classical groups 137 6 Homotopy and covering groups 146 7 Spin groups 152 8 The structure of compact groups 155 9 Representations of compact groups: Abstract theory 155 10 The Peter-Weyl theorem 158 CHAPTER VIII. The Structure of Compact Semisimple Groups 165 1 Maximal tori 165 2 The Killing form 170 3 Representations of tori 173 4 Representations of SU(2) and sl(2, C) 174 5 Roots and root spaces 177 6 Fundamental systems and their classification 183 7 Regular and singular elements 189
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