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

In this paper, we continue the analysis of [28] 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 M-a. In particular we constuct explicit Gelfand-Tsetlin bases for the representation S-n-1,S-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.