The Stanely-F\'eray-\'Sniady formula for the generalized characters of the Symmetric Group

We show that the explicit formula of Stanley-F\'eray-\'Sniady for the characters of the symmetric group have a natural extension to the generalized characters. These are the spherical functions of the unbalanced Gel'fand pair $(S_n\times S_{n-1},\text{diag}S_{n-1})$

Induced representations and harmonic analysis on finite groups

The aim of the present paper is to develop a theory of spherical functions for noncommutative Hecke algebras on finite groups. Let G be a finite group, K a subgroup and (θ, V) an irreducible, unitary K-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal basis in the commutant of IndKGV, and an associated Fourier transform. Then we translate our results in the corresponding Hecke algebra, an isomorphic algebra in the group algebra of G. Again a complete Fourier analysis is developed. As particular cases, we obtain some classical results of Curtis and Fossum on the irreducible characters. Finally, we develop a theory of Gelfand–Tsetlin bases for Hecke algebras

Harmonic analysis of finite lamplighter random walks

Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path $\mathbb{Z}$. In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the $C_2$-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. In the case the graph has a transitive isometry group $G$, we also describe the spectral analysis in terms of the representation theory of the wreath product $C_2\wr G$. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples were already studied by Haggstrom and Jonasson by probabilistic methods.Comment: 29 page

Harmonic analysis on a finite homogeneous space

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.Comment: 69 page

The discrete sine transform and the spectrum of the finite q-ary tree.

We compute the spectrum of the finite q-ary tree using radon transforms and the discrete sine transform

Fourier analysis of a class of finite radon transforms

We develop a Fourier analysis for Radon transforms between multiplicity-free permutation representations. Statistical applications of such Radon transforms were given by Diaconis and Rockmore in [ Groups and Computation ( New Brunswick, NJ, 1991), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 11, AMS, Providence, RI, 1993, pp. 87 - 104]

Multidimensional Hahn polynomials, intertwining functions on the symmetric group and Clebsch-Gordan coefficients

We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method approach for those intertwining functions. We also give a group theoretic proof of the relation between Hahn polynomials and Clebesh-Gordan coefficients, given analytically by Koornwinder and by Nikiforov, Smorodinski\u{i} and Suslov. Such relation is also extended to the multidimensional case.Comment: 31 pages, 4 figure

Multidimensional Hahn Polynomials, Intertwining Functions on the Symmetric Group and Clebsch-Gordan Coefficients

We obtain a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. We develop a tree-method approach for those intertwining functions. We give a proof of the relation between Hahn polynomials and SU(2) Clebsch-Gordan coefficients