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arXiv · math/0209042

Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials

B. Feigin·M. Jimbo·T. Miwa·E. Mukhin
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Abstract

For each pair (k,r) of positive integers with r>1, we consider an ideal I^(k,r)_n of the ring of symmetric polynomials in n variables. The ideal I_n^(k,r) has a basis consisting of Macdonald polynomials P(x_1,...,x_n;q,t) at t^{k+1}q^{r-1}=1, and is a deformed version of the one studied earlier in the context of Jack polynomials. In this paper we give a characterization of I^(k,r)_n in terms of explicit zero conditions on the k-codimensional shifted diagonals of the form x_{2}=tq^{s_1}x_1,...,x_{k+1}=tq^{s_k}x_k. The ideal I^(k,r)_n may be viewed as a deformation of the space of correlation functions of an abelian current of the affine Lie algebra \hat{sl_r}. We give a brief discussion about this connection.

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B. Feigin, M. Jimbo, T. Miwa, E. Mukhin. 2002-09-04. Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials. https://arxiv.org/abs/math/0209042

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