Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains

by Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli

Overview

Starting from a few concrete problems such as random walks on the discrete circle and the finite ultrametric space, this book develops the necessary tools for the asymptotic analysis of these processes. Its topics range from the basic theory needed for students new to this area, to advanced topics such as the theory of Green’s algebras, the complete analysis of the random matchings, and a presentation of the presentation theory of the symmetric group. This self-contained, detailed study culminates with case-by-case analyses of the cut-off phenomenon discovered by Persi Diaconis.

Product Details

ISBN-13:	9780521883368
Publisher:	Cambridge University Press
Publication date:	03/06/2008
Series:	Cambridge Studies in Advanced Mathematics , #108
Edition description:	New Edition
Pages:	454
Product dimensions:	6.30(w) x 9.21(h) x 0.98(d)

About the Author

Tullio Ceccherini-Silberstein is Professor of Mathematical Analysis in the Department of Engineering at the Università del Sannio, Benevento.

Fabio Scarabotti is Professor of Mathematical Analysis in the Department of Mathematics at the Università degli Studi di Roma 'La Sapienza'.

Filippo Tolli is Assistant Professor of Mathematical Analysis in the Department of Mathematics at the Università Roma Tre.

Table of Contents

Part I. Preliminaries, Examples and Motivations: 1. Finite Markov chains; 2. Two basic examples on Abelian groups; Part II. Representation Theory and Gelfand Pairs: 3. Basic representation theory of finite groups; 4. Finite Gelfand pairs; 5. Distance regular graphs and the Hamming scheme; 6. The Johnson Scheme and the Laplace-Bernoulli diffusion model; 7. The ultrametric space; Part III. Advanced theory: 8. Posets and the q−analogs; 9. Complements on representation theory; 10. Basic representation theory of the symmetric group; 11. The Gelfand Pair (S2n, S2 o Sn) and random matchings; Appendix 1. The discrete trigonometric transforms; Appendix 2. Solutions of the exercises; Bibliography; Index.

Editorial Reviews

"This book is, as far as I know, the first treatise fully devoted to finite Gel'fand pairs and their applications to probability and combinatories. ... It has lots of worked-out examples, and dozens of exercises (the solutions of which can be found in Appendix 2)."
Alain Valette, Mathematical Reviews

"This is the perfect group theory resource for probability theory students."
D.V. Feldman, University of New Hampshire for Choice Magazine

From the Publisher