| Description |
1 online resource (xii, 163 pages) |
| Series |
London Mathematical Society lecture note series ; 410 |
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London Mathematical Society lecture note series ; 410.
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| Contents |
880-01 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 |
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880-01/(S 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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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 |
| Summary |
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 |
| Bibliography |
Includes bibliographical references (pages 157-160) and index |
| Notes |
English |
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Print version record |
| Subject |
Harmonic analysis.
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Finite groups.
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Fourier Analysis
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MATHEMATICS -- Algebra -- Intermediate.
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Grupos finitos
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Análisis armónico
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Finite groups
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Harmonic analysis
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| Form |
Electronic book
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| Author |
Scarabotti, Fabio, author.
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Tolli, Filippo, 1968- author.
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| ISBN |
9781107732292 |
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1107732298 |
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9781107279087 |
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1107279089 |
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1139895443 |
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9781139895446 |
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1107721245 |
|
9781107721241 |
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1107730546 |
|
9781107730540 |
|
1107724171 |
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9781107724174 |
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1107728789 |
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9781107728783 |
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