On the evaluation formula for Jack polynomials with prescribed symmetry

Abstract

The Jack polynomials with prescribed symmetry are obtained from the rionsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of smallest degree, a symmetric polynomial results. Of interest in applications is the value of the latter polynomial when all the variables are set equal. Dunkl has obtained this evaluation, making use of a certain skew-symmetric operator. We introduce a simpler operator for this purpose, thereby obtaining a new derivation of the evaluation formula. An expansion formula of a certain product in terms of Jack polynomials with prescribed symmetry implied by the eval..