Representation theory and harmonic analysis of wreath products of finite groups /

Representation theory and harmonic analysis of wreath products of finite groups / Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli.

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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 formulati...

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Bibliographic Details
Online Access: Full Text (via Cambridge)
Main Authors: Ceccherini-Silberstein, Tullio (Author), Scarabotti, Fabio (Author), Tolli, Filippo, 1968- (Author)
Format: eBook
Language:English
Published: Cambridge : Cambridge University Press, 2014.
Series:London Mathematical Society lecture note series ; 410.
Subjects:
Harmonic analysis.
Finite groups.
Finite groups
Harmonic analysis
Table of Contents:
  • 1. General theory: 1.1. Induced representations; 1.1.1. Definitions; 1.1.2. Transitivity and additivity of induction; 1.1.3. Frobenius character formula; 1.1.4. Induction and restriction; 1.1.5. Induced representations and induced operators; 1.1.6. Frobenius reciprocity; 1.2. Harmonic analysis on a finite homogeneous space; 1.2.1. Frobenius reciprocity for permutation representations; 1.2.2. Spherical functions; 1.2.3. The other side of Frobenius reciprocity for permutation representations; 1.2.4. Gelfand pairs; 1.3. Clifford theory; 1.3.1. Clifford correspondence; 1.3.2. The little group method; 1.3.3. Semidirect products; 1.3.4. Semidirect products with an Abelian normal subgroup; 1.3.5. The affine group over a finite field; 1.3.6. The finite Heisenberg group
  • 2. Wreath products of finite groups and their representation theory: 2.1. Basic properties of wreath products of finite groups; 2.1.1. Definitions; 2.1.2. Composition and exponentiation actions; 2.1.3. Iterated wreath products and their actions on rooted trees; 2.1.4. Spherically homogeneous rooted trees and their automorphism group; 2.1.5. The finite ultrametric space; 2.2. Two applications of wreath products to group theory2.2.1. The theorem of Kaloujnine and Krasner; 2.2.2. Primitivity of the exponentiation action; 2.3. Conjugacy classes of wreath products; 2.3.1. A general description of conjugacy classes; 2.3.2. Conjugacy classes of groups of the form C[sub(2)] wr G; 2.3.3. Conjugacy classes of groups of the form F wr S[sub(n)]; 2.4. Representation theory of wreath products; 2.4.1. The irreducible representations of wreath products; 2.4.2. The character and matrix coefficients of the representation tilde sigma.
  • 2.5. Representation theory of groups of the form C[sub(2)] wr G2.5.1 Representation theory of the finite lamplighter group C[sub(2)] wr C[sub(n)]; 2.5.2. Representation theory of the hyperoctahedral group C[sub(2)] wr S[sub(n)]; 2.6. Representation theory of groups of the form F wr S[sub(n)]; 2.6.1. Representation theory of S[sub(m)] wr S[sub(n)]
  • 3. Harmonic analysis on some homogeneous spaces of finite wreath products: 3.1. Harmonic analysis on the composition of two permutation representations; 3.1.1. Decomposition into irreducible representations; 3.1.2. Spherical matrix coefficients; 8 3.2. The generalized Johnson scheme; 3.2.1. The Johnson scheme; 3.2.2. The homogeneous space Theta h; 3.2.3. Two special kinds of tensor product; 3.2.4. The decomposition of L (Theta [sub(h)]) into irreducible representations; 3.2.5. The spherical functions; 3.2.6. The homogeneous space V(r, s) and the associated Gelfand pair; 3.3. Harmonic analysis on exponentiations and on wreath products of permutation representations; 3.3.1. Exponentiation and wreath products; 3.3.2. The case G=C[sub(2)] and Z trivial; 3.3.3. The case when L(Y) is multiplicity free; 3.3.4. Exponentiation of finite Gelfand pairs; 3.4. Harmonic analysis on finite lamplighter spaces; 3.4.1. Finite lamplighter spaces; 3.4.2. Spectral analysis of an invariant graphs; 3.4.4. The lamplighter on the complete graph.
  • Machine generated contents note: 1. General theory
  • 1.1. Induced representations
  • 1.1.1. Definitions
  • 1.1.2. Transitivity and additivity of induction
  • 1.1.3. Frobenius character formula
  • 1.1.4. Induction and restriction
  • 1.1.5. Induced representations and induced operators
  • 1.1.6. Frobenius reciprocity
  • 1.2. Harmonic analysis on a finite homogeneous space
  • 1.2.1. Frobenius reciprocity for permutation representations
  • 1.2.2. Spherical functions
  • 1.2.3. other side of Frobenius reciprocity for permutation representations
  • 1.2.4. Gelfand pairs
  • 1.3. Clifford theory
  • 1.3.1. Clifford correspondence
  • 1.3.2. little group method
  • 1.3.3. Semidirect products
  • 1.3.4. Semidirect products with an Abelian normal subgroup
  • 1.3.5. affine group over a finite field
  • 1.3.6. finite Heisenberg group
  • 2. Wreath products of finite groups and their representation theory
  • 2.1. Basic properties of wreath products of finite groups
  • 2.1.1. Definitions
  • 2.1.2. Composition and exponentiation actions
  • 2.1.3. Iterated wreath products and their actions on rooted trees
  • 2.1.4. Spherically homogeneous rooted trees and their automorphism group
  • 2.1.5. finite ultrametric space
  • 2.2. Two applications of wreath products to group theory
  • 2.2.1. theorem of Kaloujnine and Krasner
  • 2.2.2. Primitivity of the exponentiation action
  • 2.3. Conjugacy classes of wreath products
  • 2.3.1. general description of conjugacy classes
  • 2.3.2. Conjugacy classes of groups of the form C2 G
  • 2.3.3. Conjugacy classes of groups of the form F Sn
  • 2.4. Representation theory of wreath products
  • 2.4.1. irreducible representations of wreath products
  • 2.4.2. character and matrix coefficients of the representation σ
  • 2.5. Representation theory of groups of the form C2 G
  • 2.5.1. Representation theory of the finite lamplighter group C2 Cn
  • 2.5.2. Representation theory of the hyperoctahedral group C2 Sn
  • 2.6. Representation theory of groups of the form F Sn
  • 2.6.1. Representation theory of Sm Sn
  • 3. Harmonic analysis on some homogeneous spaces of finite wreath products
  • 3.1. Harmonic analysis on the composition of two permutation representations
  • 3.1.1. Decomposition into irreducible representations
  • 3.1.2. Spherical matrix coefficients
  • 3.2. generalized Johnson scheme
  • 3.2.1. Johnson scheme
  • 3.2.2. homogeneous space h
  • 3.2.3. Two special kinds of tensor product
  • 3.2.4. decomposition of L(h) into irreducible representations
  • 3.2.5. spherical functions
  • 3.2.6. homogeneous space V(r, s) and the associated Gelfand pair
  • 3.3. Harmonic analysis on exponentiations and on wreath products of permutation representations
  • 3.3.1. Exponentiation and wreath products
  • 3.3.2. case G = C2 and Z trivial
  • 3.3.3. case when L(Y) is multiplicity free
  • 3.3.4. Exponentiation of finite Gelfand pairs
  • 3.4. Harmonic analysis on finite lamplighter spaces
  • 3.4.1. Finite lamplighter spaces
  • 3.4.2. Spectral analysis of an invariant operator
  • 3.4.3. Spectral analysis of lamplighter graphs
  • 3.4.4. lamplighter on the complete graph.

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