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

Abstract

We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group Sn and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klimyk, we develop a tree-method approach for those intertwining functions. Moreover, using our theory of $S_n$-intertwining functions and James version of the Schur- Weyl duality, we give a proof of the relation between Hahn polynomials and $SU(2)$ Clebsch-Gordan coefficients, previously obtained by Koornwinder and by Nikiforov, Smorodinskiĭ and Suslov in the $SU(2)$-setting. Such relation is also extended to the multidimensional case.