Singular Polynomials for the Symmetric Group and Krawtchouk Polynomials

Abstract

A singular polynomial is one which is annihilated by all Dunkl operators for a certain parameter value. These polynomials were first studied by Dunkl, de Jeu and Opdam, (Trans. Amer. Math. Soc. 346 (1994), 237-256). This paper constructs a family of such polynomials associated to the irreducible representation (N-2,1,1) of the symmetric group S_N for odd N and parameter values -1/2, -3/2, -5/2,... . The method depends on the use of Krawtchouk polynomials to carry out a change of variables in a generating function involved in the construction of nonsymmetric Jack polynomials labeled by (m,n,0,....), m>=n.