Abstract
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases \({t = 1}\) and \({q = 0}\), we recover known expressions for the monomial symmetric and Hall–Littlewood polynomials, respectively. Other specializations of our formula give new expressions for the Jack and q–Whittaker polynomials.
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Cantini, L., de Gier, J., Wheeler, M.: Matrix product formula for Macdonald polynomials. J. Phys. A: Math. Theor. 48, 384001 (2015). arXiv:1505.00287
Cherednik I.: Double affine Hecke algebras and Macdonald’s conjectures. Ann. Math. 141, 191–216 (1995)
Cherednik I.: Nonsymmetric Macdonald polynomials. Int. Math. Res. Not. 10, 483–515 (1995)
Duchamp, G., Krob, D., Lascoux, A., Leclerc, B., Scharf, T., Thibon, J.Y.: Euler–Poincaré characteristic and polynomial representations of Iwahori–Hecke algebras. Publ. RIMS 31, 179–201 (1995)
Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for Macdonald polynomials. J. Am. Math. Soc 18, 735–761 (2005). arXiv:math/0409538
Haglund, J., Haiman, M., Loehr, N.: A combinatorial formula for non-symmetric Macdonald polynomials. Am. J. Math. 130, 359–383 (2008). arXiv:math/0601693
Hivert F.: Hecke algebras, difference operators and quasi-symmetric functions. Adv. Math. 155, 181–238 (2000)
Kirillov, A.N., Noumi, M.: q-Difference raising operators for Macdonald polynomials and the integrality of transition coefficients. In: Algebraic Methods and q-Special Functions, CRM Proceedings and Lecture Notes, vol. 22 (1999). arXiv:q-alg/9605005
Kirillov, A.N., Noumi, M.: Affine Hecke algebras and raising operators for Macdonald polynomials. Duke Math. J. 93(1), 1–39 (1998)
Macdonald, I.: A new class of symmetric functions. Publ. I.R.M.A. Strasbourg, Actes 20\({^\textrm{e}}\) Séminaire Lotharingien, vol. 131–171 (1988)
Macdonald, I.: Symmetric functions and Hall polynomials, 2nd edn. Clarendon Press, Oxford (1995)
Opdam E.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175, 75–121 (1995)
Ram, A., Yip, M.: A combinatorial formula for Macdonald polynomials. Adv. Math. 226, 309–31 (2011) arXiv:0803.1146
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de Gier, J., Wheeler, M. A Summation Formula for Macdonald Polynomials. Lett Math Phys 106, 381–394 (2016). https://doi.org/10.1007/s11005-016-0820-3
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DOI: https://doi.org/10.1007/s11005-016-0820-3