Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials

Details Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials

2002-09-04

arxiv.org/abs/math/0209042v1

Mathematics

Quantum Algebra

Latex, 17 pages

Scientific paper

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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