  For certain negative rational numbers k0, 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 k = k0. It
was shown by de Jeu, Opdam and the author (TAMS 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. For each pair (m,n) satisfying these conditions there is a
unique irreducible S_N-module of singular polynomials for the singular value
-m/n. The existence of these polynomials was established by the author (IMRN
2004,#67,3607-3635). The uniqueness is proven in the present paper. By using
Murphy's (J. Alg. 69(1981), 287-297) results on the eigenvalues of the Murphy
elements, the problem of existence of singular polynomials is first restricted
to the isotype of a partition of N (corresponding to an irreducible
representation of S_N) such that (n/gcd(m,n)) divides t+1 for each part t of
the partition except the last one. Then by arguments involving nonsymmetric
Jack polynomials it is shown that the assumption that the second part of the
partition is greater than or equal to n/gcd(m,n) leads to a contradiction.This
shows that the singular polynomials are exactly those already determined.
