On p.118 of Harmonic Analysis on Finite Groups we read that “Persi Diaconis used Gelfand pairs in order to determine the rate of convergence to the stationary distribution of finite Markov chains. More precisely, given a Markov chain which is invariant under the action of a group G, its transition operator can be expressed as a convolution operator whose kernel can be written, at least in theory, as a ‘Fourier series’ where the classical exponentials exp(inx) are replaced by the irreducible representations of … G … This program … is not easy to handle… There are however some cases where the Fourier analysis of the action of the group over a finite set can be easily handled and reduced to an essentially commutative analysis… This is indeed the case when the action of the group on the set corresponds to a Gelfand pair…”
And what is a Gelfand pair? Well, the answer appears on p.123: it is a pair, (G,K), of finite groups, G > K, with a commutative algebra of bi-K-invariant functions. This latter object, the algebra of bi-K-invariant functions is cut out by the requirement that if g is an element of G and k, h are elements of K, then f(kgh) = f(g).
This said, it should come as no surprise that the authors of the book under review offer it as “a textbook for at least three different courses,” namely a course on finite Markov chains (Chapters 1, 2, parts of 5, 6, and Appendix 1), a course on Gelfand pairs (Chapters 1-8), or a course on the representation theory of finite groups (parts of Chapters 3, 4, then Chapters 9, 10, 11).
The book is well-crafted and thorough and the long excerpt above testifies to the authors’ clear expository style. Additionally it is particularly noteworthy for referencing and analyzing a number of evocative examples, e.g. random walk on the discrete circle and other diffusion models such as those of Ehrenfest and Bernoulli-Laplace. Also, there are a lot of useful exercises sprinkled through the text, making for a good instructional tool at the graduate level. Finally, Harmonic Analysis on Finite Groups also qualifies as a good reference for what the authors call “mature researchers,” be they group theorists, representation theorists, probabilists, or statisticians. It succeeds on many levels.
Michael Berg is Professor of Mathematics at Loyola Marymount University.
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.