Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials

2002/09/04 by B. Feigin, Feigin, B., M. Jimbo +5
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #math-ph #math.CO #math.MP #math.QA

paper · pdf · doi:10.48550/arxiv.math/0209042

Latex, 17 pages

arxiv created 2002/09/04 · arxiv updated 2009/11/30

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(x₁,...,x_n;q,t) at t^k+1q^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 x₂=tq^s₁x₁,...,x_k+1=tq^s_kx_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 sl_r. We give a brief discussion about this connection.

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