Singular Polynomials and Modules 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 de Jeu, Opdam and the author (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. For each pair (m,n) satisfying these conditions there is a unique irreducible SN module of singular polynomials for the singular value − n . The existence of these polynomials was previously 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 τ (where τ is a partition of N corresponding to an irreducible representation of SN ) satisfying the condition that n/ gcd (m,n) divides τi +1 for 1 ≤ i < l; l is the length of τ , that is, τl > τl+1 = 0. Then by arguments involving the analysis of nonsymmetric Jack polynomials it is shown that the assumption τ2 ≥ n/ gcd (m,n) leads to a contradiction. This shows that the singular polynomials are exactly those already determined, and are of isotype τ , where τ2 = . . . = τl−1 = (n/ gcd (m,n))− 1 ≥ τl.