A differential ideal of symmetric polynomials spanned by Jack polynomials at $\beta=-(r-1)/(k+1)$

Abstract

For each pair of positive integers (k,r) such that k+1,r-1 are coprime, we introduce an ideal $I^{(k,r)}_n$ of the ring of symmetric polynomials. The ideal $I^{(k,r)}_n$ has a basis consisting of Jack polynomials with parameter $\beta=-(r-1)/(k+1)$, and admits an action of a family of differential operators of Dunkl type including the positive half of the Virasoro algebra. The space $I^{(k,2)}_n$ coincides with the space of all symmetric polynomials in $n$ variables which vanish when $k+1$ variables are set equal. The space $I_n^{(2,r)}$ coincides with the space of correlation functions of an abelian current of a vertex operator algebra related to Virasoro minimal series (3,r+2).