Jack Superpolynomials with Negative Fractional Parameter: Clustering Properties and Super-Virasoro Ideals

Abstract

The Jack polynomials \({P_\lambda^{(\alpha)}}\) at α = −(k + 1)/(r − 1) indexed by certain (k, r, N)-admissible partitions are known to span an ideal \({I_{N}^{(k,r)}}\) of the space of symmetric functions in N variables. The ideal \({I_{N}^{(k,r)}}\) is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in \({I_{N}^{(k,r)}}\) admit clusters of size at most k: they vanish when k + 1 of their variables are identified, and they do not vanish when only k of them are identified. We generalize most of these properties to superspace using orthogonal eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland model known as Jack superpolynomials. In particular, we show that the Jack superpolynomials \({P_\lambda^{(\alpha)}}\) at α = −(k + 1)/(r − 1) indexed by certain (k, r, N)-admissible superpartitions span an ideal \({\mathcal{I}_{N}^{(k,r)}}\) of the space of symmetric polynomials in N commuting variables and N anticommuting variables. We prove that the ideal \({\mathcal{I}_{N}^{(k,r)}}\) is stable with respect to the action of the negative-half of the super-Virasoro algebra. In addition, we show that the Jack superpolynomials in \({\mathcal {I}_{N}^{(k,r)}}\) vanish when k + 1 of their commuting variables are equal, and conjecture that they do not vanish when only k of them are identified. This allows us to conclude that the standard Jack polynomials with prescribed symmetry should satisfy similar clustering properties. Finally, we conjecture that the elements of \({\mathcal{I}_{N}^{(k,2)}}\) provide a basis for the subspace of symmetric superpolynomials in N variables that vanish when k + 1 commuting variables are set equal to each other.