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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.

05E05 Symmetric functions and generalizations
Maple; SF
Full Text: DOI arXiv
[1] Garsia, PNAS 90 (8) pp 3607– (1993) · Zbl 0831.05062 · doi:10.1073/pnas.90.8.3607
[2] J AM MATH SOC 14 pp 941– (2001) · Zbl 1009.14001 · doi:10.1090/S0894-0347-01-00373-3
[3] PROC AM MATH SOC 121 pp 469– (1994)
[4] PROC AM MATH SOC 19 pp 236– (1968) · doi:10.1090/S0002-9939-1968-0223256-9
[5] MATH NACHR 83 pp 143– (1978) · Zbl 0319.05002 · doi:10.1002/mana.19780830111
[6] DISCRETE MATH 139 pp 219– (1995) · Zbl 0826.05056 · doi:10.1016/0012-365X(94)00134-5
[7] PROC AM MATH SOC 123 pp 2961– (1995)
[8] ELECTRON J COMBIN 5 pp R45– (1998)
[9] J SYMBOLIC COMPUT 20 pp 755– (1995) · Zbl 0849.68068 · doi:10.1006/jsco.1995.1077
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