Singular polynomials for the symmetric groups

Abstract

For certain negative rational numbers κ 0 , 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 κ=κ 0 . It was shown by the author, de Jeu, and Opdam (1994) 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. These 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 S N on irreducible spaces of singular polynomials.