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

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: June 2008
  • availability: Available
  • format: Hardback
  • isbn: 9780521883368

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  • 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.

    • Can be used as a textbook for advanced undergraduate and graduate students, and as a reference for researchers
    • First book with a complete treatment of the theory of Gelfand pairs
    • Contains 140 exercises, with solutions or generous hints, and over 60 fully-worked examples
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    Reviews & endorsements

    '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.' The Times Higher Education Supplement

    '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

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    Product details

    • Date Published: June 2008
    • format: Hardback
    • isbn: 9780521883368
    • length: 454 pages
    • dimensions: 234 x 160 x 25 mm
    • weight: 0.722kg
    • contains: 76 b/w illus. 5 tables 140 exercises
    • availability: Available
  • 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.

  • Authors

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

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

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

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