Singular polynomials for the symmetric groups

Abstract

For certain negative rational numbers k, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter equals k. It was shown by the author, de Jeu and Opdam (Trans. Amer. Math. Soc. 346 (1994), 237-256) that the singular values are exactly the values -m/n with 2 <= n <= N, m = 1,2,... and m/n is not an integer. This paper constructs for each pair (m,n) satisfying these conditions an irreducible S_N-module of singular polynomials for the singular value -m/n. The module is of isotype (n-1,(n1-1)^p,r) where n1 = n/gcd(m,n), r = N-n+1-p(n1-1) and 1 <= r <= n1-1 (thus defining p). The singular polynomials are special cases of nonsymmetric Jack polynomials. The paper presents some formulae for the action of Dunkl operators on these polynomials valid in general, and a method for showing the dependence of poles (in the parameter) on the number of variables (N). Murphy elements are used to analyze the representation of S_N on irreducible spaces of singular polynomials.