Filippo Tolli

Harmonic analysis on finite groups : representation theory, Gelfand pairs and Markov chains

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.

Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups

Preface 1. General theory 2. Wreath products of finite groups 3. Harmonic analysis on finite wreath products Bibliography Index.

Harmonic analysis on a finite homogeneous space II: The Gelfand-Tsetlin decomposition

Abstract In this paper, we continue the analysis of [Scarabotti, Tolli, Proc. London Math. Soc.: 2009] on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. We extend the theory of Gelfand–Tsetlin bases to permutation representations. Then we study several concrete examples on the symmetric groups, generalizing the Gelfand pair of the Johnson scheme. We also extend part of the Okounkov–Vershik theory to the Young permutation module Ma . In particular we constuct explicit Gelfand–Tsetlin bases for the representation S n–1,1. We also give an explicit Gelfand–Tsetlin decomposition for the permutation module associated with a three-parts partitions, using James reformulation of the Young rule by means of intertwining operators (Radon transforms). Several statistical applications, refining previous work by Diaconis, are given. Finally, the spectrum of several invariant operators is determined.

Weighted expanders and the anisotropic Alon-Boppana theorem

We present the anisotropic version of the classical Alon-Boppana theorem on the asymptotic spectrum of random walks on infinite families of graphs. The relations to the Grigorchuk-Zuk theory--on the space of graphs with uniformly bounded degree and the continuity of the spectral measure of Markov operators related to the Alon-Boppana theorem and the Burger-Greenberg-Serre theorem--are also indicated.