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Harmonic analysis on finite groups. Representation theory, Gelfand pairs and Markov chains. (English) Zbl 1149.43001
Cambridge Studies in Advanced Mathematics 108. Cambridge: Cambridge University Press (ISBN 978-0-521-88336-8/hbk). xiii, 440 p. (2008).
This text forms a self-contained introduction to Harmonic Analysis of finite Gelfand pairs and related structures, and applications to random walks on these structures where estimates for the rates of convergence of these random walks to stationary distributions in the spirit of P. Diaconis are in the center of interest. The subtitle “Representation Theory, Gelfand Pairs, and Markov Chains” of the book meets the content quite well.
The text under review is generally well-written and should be appealing to any reader interested in these algebraic structures and related random walks.
The material is presented in order of increasing difficulty and definitely not by topics in a systematic way. It seems to be a goal of the authors to present the interplay between algebraic structures and the limit behavior of random walks within of each chapter by series of examples of increasing complexity. Let us go into details of the content. In the first chapter, the authors present basic material on finite Markov chains (time homogeneous and discrete time only) and focus then on simple random walks on finite cyclic groups and on hypercubes, i.e., examples related to the Ehrenfest urn. The second chapter is devoted to basics of representation theory of finite groups (over $$\mathbb C$$) and finite Gelfand pairs as well as to application of spherical Fourier analysis to biinvariant random walks. Moreover, related structures like distance-regular graphs and association schemes are discussed. Again there is a main focus on concrete examples, here the Hamming and Johnson schemes as well as applications to the Laplace-Bernoulli diffusion model. In the final, more advanced chapter, the authors start with a detailed discussion of posets and the related Gelfand pairs and spherical functions. In particular, a computation of spherical functions via Moebius inversion due to Delsarte is included. Moreover, a short introduction to $$q$$-calculus, the $$q$$-Johnson scheme, and the nonbinary Johnson scheme are discussed. Finally, after a second general part on group representations, the representation theory of symmetric groups is developed and applied, as final highlights, to card shuffling with random transpositions due to Diaconis and Shashahani as well as to random matchings which are related with the Gelfand pair $$(S_{2n},S_2 |S_n)$$.
As mentioned above, the book under review is strongly influenced by the work of Diaconis (and his collaborators) and in particular by his well-known lucid booklet on group representations in probability and statistics. Moreover, partially there are close connections to the project of Aldous and Fill and the textbook of Behrends on Markov chains, the monographs of Bannai and Ito and of Bailey on association schemes, the booklet of Krieg on Hecke algebras, the book of Terras on Harmonic Analysis of finite groups, and many other texts. However, possibly up to the booklet of Diaconis, there is no textbook covering precisely the main focus of the book under review. For this reason, and as the text really forms a completely self-contained introduction to this interesting field on the edge between algebra and probability, I recommend the text to everybody interested in this topic. I also expect that it will serve as a good source for seminars.
In the eyes of the reviewer, rather a probabilist than an algebraist, the book has only one disadvantage, namely that the focus is too much on some underlying algebraic structures and a few exclusive examples only. For instance, the part on more advanced, but classical $$q$$-examples is quite short. Moreover, many other closely related topics are excluded almost completely (some subjects at least appear in the bibliography, but without any reference within the text). In particular, many further finite examples, infinite Gelfand pairs (even discrete and easy ones like infinite distance-transitive graphs) and any discussion of spherical functions as finite orthogonal polynomials within the theory of special functions do not appear at all. Moreover, the concept of association schemes is quite complicated, if one is interested in random walks on distance-regular graphs only. The reviewer would prefer in this context the notion of finite commutative hypergroups (in the sense of Dunkl, Jewett, and Spector), as this structure fits much better to random walks on distance-regular graphs.
Moreover, this point of view sometimes would lead to more general results without additional input. For instance, the Ehrenfest model may be regarded as Markov chain on $$\{0,\ldots,N\}$$ with the discrete parameter $$N$$ and with a second continuous parameter $$p\in]0,1[$$ with a concrete meaning. Furthermore, the convergence of the Krawtchouk polynomials, which form the associated spherical functions here, to Hermite polynomials is closely related with the convergence of the Ehrenfest Markov chains to the Ornstein-Uhlenbeck process where this central limit theorem explains the cutoff estimates very precisely.
It is clear, that it is not possible to write an introductory, completely self-contained text that deals with all related topics, but the complete lack of discussion of many topics suggests in the eyes of the reviewer too much that the subject is closed, and that affirmative answers are given mainly by algebraic methods, which is definitely not the case. Therefore, the text is a good and very useful introduction driven by particular examples, but not a research monograph.

##### MSC:
 43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis 43A05 Measures on groups and semigroups, etc. 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 43A90 Harmonic analysis and spherical functions 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 05E30 Association schemes, strongly regular graphs