B. Feigin, M. Jimbo, T. Miwa, E. Mukhin; Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials. Int Math Res Notices 2003; 2003 (18): 1015-1034. doi: 10.1155/S1073792803209119
For each pair (k, r) of positive integers with r ≥ 2, we consider an ideal of the ring of symmetric polynomials. The ideal 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 terms of explicit zero conditions on the k-codimensional shifted diagonals of the form . The ideal may be viewed as a deformation of the space of correlation functions of an abelian current of the affine Lie algebra . We give a brief discussion about this connection.