A combinatorial model for the Macdonald polynomials

Abstract

We introduce a polynomial C̃ μ [ Z ; q, t ], depending on a set of variables Z = z 1 , z 2 ,..., a partition μ, and two extra parameters q, t . The definition of C̃ μ involves a pair of statistics (maj(σ, μ), inv(σ, μ)) on words σ of positive integers, and the coefficients of the z i are manifestly in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathbb{N}}[q,t]\end{equation*}\end{document} . We conjecture that C̃ μ [ Z ; q, t ] is none other than the modified Macdonald polynomial H̃ μ [ Z ; q, t ]. We further introduce a general family of polynomials F T [ Z ; q, S ], where T is an arbitrary set of squares in the first quadrant of the xy plane, and S is an arbitrary subset of T . The coefficients of the F T [ Z ; q, S ] are in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\mathbb{N}}[q]\end{equation*}\end{document} , and C̃ μ [ Z ; q, t ] is a sum of certain F T [ Z ; q, S ] times nonnegative powers of t . We prove F T [ Z ; q, S ] is symmetric in the z i and satisfies other properties consistent with the conjecture. We also show how the coefficient of a monomial in F T [ Z ; q, S ] can be expressed recursively. maple calculations indicate the F T [ Z ; q, S ] are Schur-positive, and we present a combinatorial conjecture for their Schur coefficients when the set T is a partition with at most three columns.