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Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups

Book Description

This book presents an introduction to the representation theory of wreath products of finite groups and harmonic analysis on the corresponding homogeneous spaces. The reader will find a detailed description of the theory of induced representations and Clifford theory, focusing on a general formulation of the little group method. This provides essential tools for the determination of all irreducible representations of wreath products of finite groups. The exposition also includes a detailed harmonic analysis of the finite lamplighter groups, the hyperoctahedral groups, and the wreath product of two symmetric groups. This relies on the generalised Johnson scheme, a new construction of finite Gelfand pairs. The exposition is completely self-contained and accessible to anyone with a basic knowledge of representation theory. Plenty of worked examples and several exercises are provided, making this volume an ideal textbook for graduate students. It also represents a useful reference for more experienced researchers.

Table of Contents

  1. Cover
  2. Series
  3. Title
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. 1 General theory
    1. 1.1 Induced representations
      1. 1.1.1 Definitions
      2. 1.1.2 Transitivity and additivity of induction
      3. 1.1.3 Frobenius character formula
      4. 1.1.4 Induction and restriction
      5. 1.1.5 Induced representations and induced operators
      6. 1.1.6 Frobenius reciprocity
    2. 1.2 Harmonic analysis on a finite homogeneous space
      1. 1.2.1 Frobenius reciprocity for permutation representations
      2. 1.2.2 Spherical functions
      3. 1.2.3 The other side of Frobenius reciprocity for permutation representations
      4. 1.2.4 Gelfand pairs
    3. 1.3 Clifford theory
      1. 1.3.1 Clifford correspondence
      2. 1.3.2 The little group method
      3. 1.3.3 Semidirect products
      4. 1.3.4 Semidirect products with an Abelian normal subgroup
      5. 1.3.5 The affine group over a finite field
      6. 1.3.6 The finite Heisenberg group
  9. 2 Wreath products of finite groups and their representation theory
    1. 2.1 Basic properties of wreath products of finite groups
      1. 2.1.1 Definitions
      2. 2.1.2 Composition and exponentiation actions
      3. 2.1.3 Iterated wreath products and their actions on rooted trees
      4. 2.1.4 Spherically homogeneous rooted trees and their automorphism group
      5. 2.1.5 The finite ultrametric space
    2. 2.2 Two applications of wreath products to group theory
      1. 2.2.1 The theorem of Kaloujnine and Krasner
      2. 2.2.2 Primitivity of the exponentiation action
    3. 2.3 Conjugacy classes of wreath products
      1. 2.3.1 A general description of conjugacy classes
      2. 2.3.2 Conjugacy classes of groups of the form C[sub(2)] ≀ G
      3. 2.3.3 Conjugacy classes of groups of the form F ≀ S[sub(n)]n
    4. 2.4 Representation theory of wreath products
      1. 2.4.1 The irreducible representations of wreath products
      2. 2.4.2 The character and matrix coefficients of the representation σ
    5. 2.5 Representation theory of groups of the form C[sub(2)] ≀ G
      1. 2.5.1 Representation theory of the finite lamplighter group C[sub(2)] ≀ C[sub(n)]
      2. 2.5.2 Representation theory of the hyperoctahedral group C[sub(2)] ≀ S[sub(n)]
    6. 2.6 Representation theory of groups of the form F ≀ S[sub(n)]
      1. 2.6.1 Representation theory of S[sub(m)] ≀ S[sub(n)]
  10. 3 Harmonic analysis on some homogeneous spaces of finite wreath products
    1. 3.1 Harmonic analysis on the composition of two permutation representations
      1. 3.1.1 Decomposition into irreducible representations
      2. 3.1.2 Spherical matrix coefficients
    2. 3.2 The generalized Johnson scheme
      1. 3.2.1 The Johnson scheme
      2. 3.2.2 The homogeneous space Θ[sub(h)]
      3. 3.2.3 Two special kinds of tensor product
      4. 3.2.4 The decomposition of L(Θ[sub(h)]) into irreducible representations
      5. 3.2.5 The spherical functions
      6. 3.2.6 The homogeneous space V(r, s) and the associated Gelfand pair
    3. 3.3 Harmonic analysis on exponentiations and on wreath products of permutation representations
      1. 3.3.1 Exponentiation and wreath products
      2. 3.3.2 The case G = C[sub(2)] and Z trivial
      3. 3.3.3 The case when L(Y ) is multiplicity free
      4. 3.3.4 Exponentiation of finite Gelfand pairs
    4. 3.4 Harmonic analysis on finite lamplighter spaces
      1. 3.4.1 Finite lamplighter spaces
      2. 3.4.2 Spectral analysis of an invariant operator
      3. 3.4.3 Spectral analysis of lamplighter graphs
      4. 3.4.4 The lamplighter on the complete graph
  11. References
  12. Index