Harmonic Analysis on Finite Groups
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, written by three Italian mathematicians, is a rare example. Its main aim is to introduce some modern tools for studying asymptotic behaviour of Markov chains on certain finite groups. This goal is illustrated in the very beginning on four examples: a random walk on a circle, the Ehrenfest and Bernoulli-Laplace diffusion models and a Markov chain on the group of permutations defined via random transpositions. Asymptotics of some of these models shows a surprising cut-off phenomenon, or a sudden transition from "order" to "chaos" after a certain (large) number of steps. This and other phenomena are studied mostly with the tools of representation theory of finite groups. These are introduced starting from the most elementary level, with complete proofs, many examples and exercises with solutions, so that the book would be perfectly suitable for an introductory course on discrete Fourier transforms and related representation theory. 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. These are pairs of a group G and a subgroup K such that the decomposition of the set of functions on the homogeneous space G/K is multiplicity free. The Gelfand pair action on a Markov chain, if available, often simplifies the analysis of its asymptotic behaviour. This is illustrated in detail on the examples introduced at the beginning. There is an extensive bibliography and an appendix on discrete trigonometric transforms. The book can be warmly recommended for anyone interested in the subject and/or looking for interesting applications of representation theory.