Singular Polynomials for the Symmetric Groups

For certain negative rational numbers κ0, called singular values, and associated with the symmetric group SN on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter κ = κ0. 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 − 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 SN -module of singular polynomials for the singular value − n . The module is of isotype ( n− 1, (n1 − 1) l , ρ ) where n1 = n/ gcd(m,n), ρ = N − (n− 1) − l (n1 − 1) and 1 ≤ ρ ≤ n1 − 1. 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 κ) on the number of variables. Murphy elements are used to analyze the representation of SN on irreducible spaces of singular polynomials.