Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals

Details Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals Jack superpolynomials with negative fractional parameter: clustering properties and super-Virasoro ideals

2011-09-13

arxiv.org/abs/1109.2832v1

Physics

Mathematical Physics

34 pages

Scientific paper

The Jack polynomials $P_\la^{(\a)}$ at $\a=-(k+1)/(r-1)$ indexed by certain $(k,r,N)$-admissible partitions are known to span an ideal $I^{(k,r)}_N$ of the space of symmetric functions in $N$ variables. The ideal $I^{(k,r)}_N$ is invariant under the action of certain differential operators which include half the Virasoro algebra. Moreover, the Jack polynomials in $I^{(k,r)}_N$ 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 $\a=-(k+1)/(r-1)$ indexed by certain $(k,r,N)$-admissible superpartitions span an ideal ${\mathcal I}^{(k,r)}_N$ of the space of symmetric polynomials in $N$ commuting variables and $N$ anticommuting variables. We prove that the ideal ${\mathcal I}^{(k,r)}_N$ is stable with respect to the action of the negative-half of the super-Virasoro algebra. Finally, we show that the Jack superpolynomials in ${\mathcal I}^{(k,r)}_N$ 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.

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