The present chapter is an exposition of the results in the papers [9, 63, 64]. We have rearranged material in the original sources, adding more details and making everything consistent with the background developed in the preceding chapters. In the first section we examine the composition of two permutation representations (possibly with multiplicities, that is, not necessarily yielding Gelfand pairs) and present the corresponding decomposition into irreducible subrepresentations. We also give an explicit expression for the associated spherical matrix coefficients. In Section 3.2 we study the generalized Johnson scheme and describe a general construction of finite Gelfand pairs, introduced in [9], which is based on the action of the group Aut(T) of automorphisms of a finite rooted tree T on the space of all rooted subtrees of T. We then study the harmonic analysis of the exponentiation, following [64], and analyze in detail the lamplighter group by developing a harmonic analysis on the corresponding finite lamplighter spaces.

Harmonic analysis on the composition of two permutation representations

In this section we examine the composition of two permutation representations. We give the rule for decomposition into irreducible representations and the formulas for the related spherical matrix coefficients. For a more general treatment, namely for the harmonic analysis on the composition action of a crested product (a generalization of both the direct product and the wreath product), we refer to [64].