  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.

Representation Theory;Quantum Algebra