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A combinatorial model for the Macdonald polynomials. (English) Zbl 1064.05147
Summary: We introduce a polynomial \(\overline{C}_\mu[Z;q,t]\), depending on a set of variables \(Z= \{z_1, \dots\}\), a partition \(\mu\), and two extra parameters \(q\), \(t\). The definition of \(\overline{C}_\mu\) involves a pair of statistics \((\text{maj}(\sigma,\mu), \text{inv}(\sigma,\mu))\) on words \(\sigma\) of positive integers, and the coefficients of the \(z_i\) are manifestly in \(\mathbb N[q,t]\). We conjecture that \(\overline{C}_\mu[Z;q,t]\) is none other than the modified Macdonald polynomial \(\overline{H}_\mu[Z; q, t]\). 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 \(\mathbb N[q]\), and \(\overline{C}_\mu[Z;q,t]\) 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.

MSC:
05E05 Symmetric functions and generalizations
Software:
Maple; SF
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