CONTENTS ix 8 The Weyl group 192 9 The classical groups 196 CHAPTER IX. The Representations of Compact Semisimple Groups 205 1 Geometry of the Cartan-Stiefel diagram 206 2 The geometry of integral forms 210 3 The Weyl integration formula 213 4 Maximal weights 215 5 The classification theorem and the Weyl character formula 217 6 Consequences of the Weyl character formula 219 7 Representation theory: The algebraic approach 225 8 Representations of the classical groups 227 9 Determinant formulas for the classical characters 237 10 Real and quaternionic representations of the classical groups 242 11 Tensors, permutations, and the Frobenius character formula 246 Appendices A Multilinear algebra 253 B The analysis of self-adjoint Hilbert-Schmidt operators 257 Bibliography 261 Index 263

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