A combinatorial model for the Macdonald polynomials
AUTOR(ES)
Haglund, J.
FONTE
National Academy of Sciences
RESUMO
We introduce a polynomial C̃μ[Z; q, t], depending on a set of variables Z = z1, z2,..., a partition μ, and two extra parameters q, t. The definition of C̃μ involves a pair of statistics (maj(σ, μ), inv(σ, μ)) on words σ of positive integers, and the coefficients of the zi 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 C̃μ[Z; q, t] is none other than the modified Macdonald polynomial H̃μ[Z; q, t]. We further introduce a general family of polynomials FT[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 FT[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 C̃μ[Z; q, t] is a sum of certain FT[Z; q, S] times nonnegative powers of t. We prove FT[Z; q, S] is symmetric in the zi and satisfies other properties consistent with the conjecture. We also show how the coefficient of a monomial in FT[Z; q, S] can be expressed recursively. maple calculations indicate the FT[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.
