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El. knyga: Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups

(Universitą degli Studi del Sannio, Italy), (Universitą degli Studi di Roma 'La Sapienza', Italy), (Universitą degli Studi Roma Tre)
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Representation Theory and Harmonic Analysis of Wreath Products of Finite GroupsDidesnis paveikslėlis
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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 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.

A self-contained treatment of the representation theory of wreath products of finite groups and harmonic analysis on the corresponding homogeneous spaces. The book functions as both a useful reference for researchers, and a graduate textbook with plenty of examples and several exercises.

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A self-contained introduction to the representation theory and harmonic analysis of wreath products of finite groups, with examples and exercises.
Preface xi
1 General theory
1(59)
1.1 Induced representations
1(15)
1.1.1 Definitions
1(7)
1.1.2 Transitivity and additivity of induction
8(2)
1.1.3 Frobenius character formula
10(1)
1.1.4 Induction and restriction
11(3)
1.1.5 Induced representations and induced operators
14(1)
1.1.6 Frobenius reciprocity
14(2)
1.2 Harmonic analysis on a finite homogeneous space
16(25)
1.2.1 Frobenius reciprocity for permutation representations
16(6)
1.2.2 Spherical functions
22(12)
1.2.3 The other side of Frobenius reciprocity for permutation representations
34(3)
1.2.4 Gelfand pairs
37(4)
1.3 Clifford theory
41(19)
1.3.1 Clifford correspondence
42(7)
1.3.2 The little group method
49(1)
1.3.3 Semidirect products
50(1)
1.3.4 Semidirect products with an Abelian normal subgroup
51(1)
1.3.5 The affine group over a finite field
52(4)
1.3.6 The finite Heisenberg group
56(4)
2 Wreath products of finite groups and their representation theory
60(44)
2.1 Basic properties of wreath products of finite groups
60(16)
2.1.1 Definitions
60(3)
2.1.2 Composition and exponentiation actions
63(4)
2.1.3 Iterated wreath products and their actions on rooted trees
67(2)
2.1.4 Spherically homogeneous rooted trees and their automorphism group
69(1)
2.1.5 The finite ultrametric space
70(6)
2.2 Two applications of wreath products to group theory
76(5)
2.2.1 The theorem of Kaloujnine and Krasner
76(2)
2.2.2 Primitivity of the exponentiation action
78(3)
2.3 Conjugacy classes of wreath products
81(11)
2.3.1 A general description of conjugacy classes
82(4)
2.3.2 Conjugacy classes of groups of the form C2 G
86(3)
2.3.3 Conjugacy classes of groups of the form F Sn
89(3)
2.4 Representation theory of wreath products
92(6)
2.4.1 The irreducible representations of wreath products
92(3)
2.4.2 The character and matrix coefficients of the representation σ
95(3)
2.5 Representation theory of groups of the form C2 G
98(3)
2.5.1 Representation theory of the finite lamplighter group C2 Cn
99(1)
2.5.2 Representation theory of the hyperoctahedral group C2 Sn
100(1)
2.6 Representation theory of groups of the form F Sn
101(3)
2.6.1 Representation theory of Sm Sn
103(1)
3 Harmonic analysis on some homogeneous spaces of finite wreath products
104(53)
3.1 Harmonic analysis on the composition of two permutation representations
104(6)
3.1.1 Decomposition into irreducible representations
104(3)
3.1.2 Spherical matrix coefficients
107(3)
3.2 The generalized Johnson scheme
110(20)
3.2.1 The Johnson scheme
110(2)
3.2.2 The homogeneous space h
112(5)
3.2.3 Two special kinds of tensor product
117(3)
3.2.4 The decomposition of L(h) into irreducible representations
120(3)
3.2.5 The spherical functions
123(4)
3.2.6 The homogeneous space V(r, s) and the associated Gelfand pair
127(3)
3.3 Harmonic analysis on exponentiations and on wreath products of permutation representations
130(15)
3.3.1 Exponentiation and wreath products
130(9)
3.3.2 The case G = C2 and Z trivial
139(3)
3.3.3 The case when L(Y) is multiplicity free
142(2)
3.3.4 Exponentiation of finite Gelfand pairs
144(1)
3.4 Harmonic analysis on finite lamplighter spaces
145(12)
3.4.1 Finite lamplighter spaces
145(3)
3.4.2 Spectral analysis of an invariant operator
148(2)
3.4.3 Spectral analysis of lamplighter graphs
150(3)
3.4.4 The lamplighter on the complete graph
153(4)
References 157(4)
Index 161
Tullio Ceccherini-Silberstein is a Professor at the Universitą del Sannio, Italy. Fabio Scarabotti is a Professor at Sapienza Universitą di Roma, Italy. Filippo Tolli is a Professor at Universitą Roma Tre, Italy.
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