Abstract

For each pair (k, r) of positive integers with r ≥ 2, we consider an ideal In(k,r) of the ring of symmetric polynomials. The ideal In(k,r) has a basis consisting of Macdonald polynomials Pλ(x1,…,xn;q,t) at tk+1qr−1=1, and is a deformed version of the one studied earlier by the authors (2002). In this paper, we give a characterization of In(k,r) in terms of explicit zero conditions on the k-codimensional shifted diagonals of the form x2=tqs1x1,,xk+1=tqskxk. The ideal In(k,r) may be viewed as a deformation of the space of correlation functions of an abelian current of the affine Lie algebra s^r. We give a brief discussion about this connection.

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