1 Introduction 2 Review of Linear Algebra 2.1 Basic Definitions and Notation 2.2 Complex Inner Product Spaces 2.3 Further Notions from Linear Algebra 3 Group Representations 3.1 Basic Definitions and First Examples 3.2 Maschke's Theorem and Complete Reducibility 4 Character Theory and the Orthogonality Relations 4.1 Morphisms of Representations 4.2 The Orthogonality Relations 4.3 Characters and Class Functions 4.4 The Regular Representation 4.5 Representations of Abelian Groups 5 Fourier Analysis on Finite Groups 5.1 Periodic Functions on Cyclic Groups 5.2 The Convolution Product 5.3 Fourier Analysis on Finite Abelian Groups 5.4 An Application to Graph Theory 5.5 Fourier Analysis on Non-abelian Groups 6 Burnside's Theorem 6.1 A Little Number Theory 6.2 The Dimension Theorem 6.3 Burnside's Theorem 7 Group Actions and Permutation Representations 7.1 Group Actions 7.2 Permutation Representations 7.3 The Centralizer Algebra and Gelfand Pairs 8 Induced Representations 8.1 Induced Characters and Frobenius Reciprocity 8.2 Induced Representations 8.3 Mackey's Irreducibility Criterion 9 Another Theorem of Burnside 9.1 Conjugate Representations 10 Representation Theory of the Symmetric Group 10.1 Partitions and Tableaux 10.2 Constructing the Irreducible Representations 11 Probability and Random Walks on Groups 11.1 Probabilities on Groups 11.2 Random Waks on Finite Groups 11.3 Card Shuffling 11.3.1 The Riffle Shuffle 11.4 The Spectrum and the Upper Bound Lemma References Index
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