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

Self-contained text, ideal for graduate students new to the area and researchers.

Tullio Ceccherini-Silberstein (Author), Fabio Scarabotti (Author), Filippo Tolli (Author)

9780521883368, Cambridge University Press

Hardback, published 6 March 2008

454 pages, 76 b/w illus. 5 tables 140 exercises
23.4 x 16 x 2.5 cm, 0.722 kg

'There are not many books that can be used both as an elementary textbook and a research monograph with the same ease and success. This one … is a rare example. … No prerequisites on probability theory and Markov chains are required; everything is explained in detail. From a researcher's point of view, the introduction and detailed study of Gelfand pairs in the context of finite groups is very valuable. … The book can be warmly recommended for anyone interested in the subject and/or looking for interesting applications of representation theory.' EMS Newsletter

Line up a deck of 52 cards on a table. Randomly choose two cards and switch them. How many switches are needed in order to mix up the deck? 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. This detailed study culminates with the case-by-case analysis of the cut-off phenomenon discovered by Persi Diaconis. This self-contained text is ideal for graduate students and researchers working in the areas of representation theory, group theory, harmonic analysis and Markov chains. 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 the representation theory of the symmetric group.

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.

Subject Areas: Calculus & mathematical analysis [PBK], Algebra [PBF]