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

International Mathematics Research Notices. 2002. No. 23. P. 1223-1237.
Feigin B. L., Jimbo M., Miwa T., Mukhin E.

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 C[x1, · · ·, xn]Sn.
The ideal I(k,r) n has a basis consisting of Jack polynomials with parameter = −(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(2,r)n coincides with the space of correlation functions of an abelian current of a vertex operator algebra related to Virasoro minimal series (3, r + 2).