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 . 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 , 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.