Comparing formulas for type GL n Macdonald polynomials
Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 849-883.

The paper compares (and reproves) the alcove walk and the nonattacking fillings formulas for type GL n Macdonald polynomials which were given in [10], [1] and [18]. The “compression” relating the two formulas in this paper is the same as that of Lenart [13]. We have reformulated it so that it holds without conditions and so that the proofs of the alcove walks formula and the nonattacking fillings formula are parallel. This reformulation highlights the role of the double affine Hecke algebra and Cherednik’s intertwiners. An exposition of the type GL n double affine braid group, double affine Hecke algebra, and all definitions and proofs regarding Macdonald polynomials are provided to make this paper self contained.

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Accepted:
Published online:
DOI: 10.5802/alco.227
Classification: 05E05,  33D52
Keywords: Macdonald polynomials, affine Hecke algebras, tableaux
Guo, Weiying 1; Ram, Arun 1

1 University of Melbourne School of Mathematics and statistics Parkville Vic 3010. Melbourne
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Guo, Weiying; Ram, Arun. Comparing formulas for type $GL_n$ Macdonald polynomials. Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 849-883. doi : 10.5802/alco.227. https://alco.centre-mersenne.org/articles/10.5802/alco.227/

[1] Alexandersson, Per Non-symmetric Macdonald polynomials and Demazure-Lusztig operators, Sém. Lothar. Combin., Volume 76 ([2016-2019]), Paper no. B76d, 27 pages | MR

[2] Borodin, Alexei; Wheeler, Michael Nonsymmetric Macdonald polynomials via integrable vertex models (2019) | arXiv

[3] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, xii+300 pages (Translated from the 1968 French original by Andrew Pressley) | DOI | MR

[4] Cherednik, Ivan Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices (1995) no. 10, pp. 483-515 | DOI | MR | Zbl

[5] Cherednik, Ivan Double affine Hecke algebras, London Mathematical Society Lecture Note Series, 319, Cambridge University Press, Cambridge, 2005, xii+434 pages | DOI | MR

[6] Corteel, Sylvie; Mandelshtam, Olya; Williams, Lauren From multiline queues to Macdonald polynomials via the exclusion process, Sém. Lothar. Combin., Volume 82B (2020), Paper no. 97, 12 pages | MR | Zbl

[7] Feigin, Evgeny; Makedonskyi, Ievgen; Orr, Daniel Generalized Weyl modules and nonsymmetric q-Whittaker functions, Adv. Math., Volume 330 (2018), pp. 997-1033 | DOI | MR | Zbl

[8] Ferreira, Jeffrey Paul Row-strict Quasisymmetric Schur Functions, Characterizations of Demazure Atoms, and Permuted Basement Nonsymmetric Macdonald Polynomials, ProQuest LLC, Ann Arbor, MI, 2011, 90 pages Thesis (Ph.D.)–University of California, Davis | MR

[9] Guo, Weiying; Ram, Arun Comparing formulas for type GL n Macdonald polynomials – Supplement, Algebr. Comb., Volume 5 (2022) no. 5, pp. 885-923 | arXiv

[10] Haglund, J.; Haiman, M.; Loehr, N. A combinatorial formula for nonsymmetric Macdonald polynomials, Amer. J. Math., Volume 130 (2008) no. 2, pp. 359-383 | DOI | MR | Zbl

[11] Knop, Friedrich Integrality of two variable Kostka functions, J. Reine Angew. Math., Volume 482 (1997), pp. 177-189 | DOI | MR | Zbl

[12] Knop, Friedrich; Sahi, Siddhartha A recursion and a combinatorial formula for Jack polynomials, Invent. Math., Volume 128 (1997) no. 1, pp. 9-22 | DOI | MR | Zbl

[13] Lenart, Cristian On combinatorial formulas for Macdonald polynomials, Adv. Math., Volume 220 (2009) no. 1, pp. 324-340 | DOI | MR | Zbl

[14] Macdonald, I. G. Symmetric functions and Hall polynomials. With contributions by A. Zelevinsky, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995, x+475 pages | MR

[15] Macdonald, I. G. Affine Hecke algebras and orthogonal polynomials, Astérisque (1996) no. 237, pp. Exp. No. 797, 4, 189-207 (Séminaire Bourbaki, Vol. 1994/95) | Numdam | MR | Zbl

[16] Macdonald, I. G. Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157, Cambridge University Press, Cambridge, 2003, x+175 pages | DOI | MR

[17] Ram, Arun Affine Hecke algebras and generalized standard Young tableaux, J. Algebra, Volume 260 (2003) no. 1, pp. 367-415 (Special issue celebrating the 80th birthday of Robert Steinberg) | DOI | MR | Zbl

[18] Ram, Arun; Yip, Martha A combinatorial formula for Macdonald polynomials, Adv. Math., Volume 226 (2011) no. 1, pp. 309-331 | DOI | MR | Zbl

[19] Sahi, Siddhartha Interpolation, integrality, and a generalization of Macdonald’s polynomials, Internat. Math. Res. Notices (1996) no. 10, pp. 457-471 | DOI | MR | Zbl

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