  In this paper, we study harmonic analysis on finite homogeneous spaces whose
associated permutation representation decomposes with multiplicity. After a
careful look at Frobenius reciprocity and transitivity of induction, and the
introduction of three types of spherical functions, we develop a theory of
Gelfand Tsetlin bases for permutation representations. Then we study several
concrete examples on the symmetric groups, generalizing the Gelfand pair of the
Johnson scheme; we also consider statistical and probabilistic applications.
After that, we consider the composition of two permutation representations,
giving a non commutative generalization of the Gelfand pair associated to the
ultrametric space; actually, we study the more general notion of crested
product. Finally, we consider the exponentiation action, generalizing the
decomposition of the Gelfand pair of the Hamming scheme; actually, we study a
more general construction that we call wreath product of permutation
representations, suggested by the study of finite lamplighter random walks. We
give several examples of concrete decompositions of permutation representations
and several explicit 'rules' of decomposition.
