Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups

Abstract

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

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the categoryF≀FIn

“…It will be often convenient to draw the diagram λ instead of writing S λ . For instance we may write ⊕ instead of: S λ ⊕ S δ for partitions λ = [3] and δ = [2, 1].…”

“…The most classical case is where H = S n−1 viewed as the subgroup of all permutations that fix n. An important generalization is the Littlewood-Richardson rule which gives the answer for the case H = S k × S n−k .Let F and G be finite groups such that G acts on the left of a finite set X. We denote by F ≀ X G the wreath product of F and G. The representation theory of F ≀ X G is a well-studied subject (see [3] and [6, Chapter 4]) and the case G = S n with the natural action on {1, . .…”

“…The Littlewood-Richardson coefficient c γ λ,δ is the number of semi-standard skew tableaux of shape γ/λ with content δ whose row word is a lattice permutation. If λ = [2, 1], δ =[3,2] and γ = [4, 3, 1] then c γ λ,δ = 2 since there are two skew tableaux with the required properties. These are:…”

The set of stable primes for polynomial sequences with large Galois group

“…Note that thanks to the separability assumption, every vertex at level n has exactly d n+1 descendants at level n + 1. Such a tree is called spherically homogeneous rooted tree (see for example [1] and [5]), and it depends only on the sequence of the degrees (d 1 , d 2 , . .…”

Wreath Product Action on Generalized Boolean Algebras

“…This generalizes the case of the Johnson scheme [4] (corresponding to |X| = 1) and the nonbinary Johnson scheme [13] (corresponding to G = group of all permutations of X). For other examples of this framework see [2,3]. We consider the related problem of explicitly block diagonalizing the commutant A X (n) of the action of G ∼ S n on V (B X (n)).…”