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 $I_n^{(k,r)}$ of the ring of symmetric polynomials. The ideal $I_n^{(k,r)}$ 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 $I_n^{(k,2)}$ 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).

Latex, 12 pages