Polynomial bases: Positivity and Schur multiplication

Searles, Dominic

doi:10.1090/tran/7670

Polynomial bases: Positivity and Schur multiplication
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by Dominic Searles PDF

Trans. Amer. Math. Soc. 373 (2020), 819-847 Request permission

Abstract:

We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters, and Demazure atoms; the quasi-key, fundamental, and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and that an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial. This rule simultaneously extends Haglund, Luoto, Mason, and van Willigenburg’s (2011) Littlewood-Richardson rule for quasi-Schur polynomials and refines their Littlewood-Richardson rule for Demazure characters. We also establish bijections connecting combinatorial models for these polynomials including semi-skyline fillings and quasi-key tableaux.

References

Per Alexandersson, Non-symmetric Macdonald polynomials and Demazure-Lusztig operators, preprint, arXiv:1602.05153 (2016).
Sami Assaf and Dominic Searles, Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams, Adv. Math. 306 (2017), 89–122. MR 3581299, DOI 10.1016/j.aim.2016.10.015
Sami Assaf and Dominic Searles, Kohnert tableaux and a lifting of quasi-Schur functions, J. Combin. Theory Ser. A 156 (2018), 85–118. MR 3762104, DOI 10.1016/j.jcta.2018.01.001
Sami Assaf, Combinatorial models for Schubert polynomials, preprint (2017), arXiv:1703.00088.
Sami Assaf, Weak dual equivalence for polynomials, preprint, arXiv:1702.04051 (2017).
Sami Assaf, Nonsymmetric Macdonald polynomials and a refinement of Kostka-Foulkes polynomials, Trans. Amer. Math. Soc. 370 (2018), no. 12, 8777–8796. MR 3864395, DOI 10.1090/tran/7374
Sara C. Billey, William Jockusch, and Richard P. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), no. 4, 345–374. MR 1241505, DOI 10.1023/A:1022419800503
Ivan Cherednik, Nonsymmetric Macdonald polynomials, Internat. Math. Res. Notices 10 (1995), 483–515. MR 1358032, DOI 10.1155/S1073792895000341
Michel Demazure, Une nouvelle formule des caractères, Bull. Sci. Math. (2) 98 (1974), no. 3, 163–172. MR 430001
Ira M. Gessel, Multipartite $P$-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289–317. MR 777705, DOI 10.1090/conm/034/777705
J. Haglund, M. Haiman, and N. Loehr, A combinatorial formula for nonsymmetric Macdonald polynomials, Amer. J. Math. 130 (2008), no. 2, 359–383. MR 2405160, DOI 10.1353/ajm.2008.0015
J. Haglund, K. Luoto, S. Mason, and S. van Willigenburg, Quasisymmetric Schur functions, J. Combin. Theory Ser. A 118 (2011), no. 2, 463–490. MR 2739497, DOI 10.1016/j.jcta.2009.11.002
J. Haglund, K. Luoto, S. Mason, and S. van Willigenburg, Refinements of the Littlewood-Richardson rule, Trans. Amer. Math. Soc. 363 (2011), no. 3, 1665–1686. MR 2737282, DOI 10.1090/S0002-9947-2010-05244-4
Bogdan Ion, Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J. 116 (2003), no. 2, 299–318. MR 1953294, DOI 10.1215/S0012-7094-03-11624-5
Axel Kohnert, Weintrauben, Polynome, Tableaux, Bayreuth. Math. Schr. 38 (1991), 1–97 (German). Dissertation, Universität Bayreuth, Bayreuth, 1990. MR 1132534
Allen Knutson and Alexander Yong, A formula for $K$-theory truncation Schubert calculus, Int. Math. Res. Not. 70 (2004), 3741–3756. MR 2101981, DOI 10.1155/S1073792804142244
Thomas Lam and Pavlo Pylyavskyy, Combinatorial Hopf algebras and $K$-homology of Grassmannians, Int. Math. Res. Not. IMRN 24 (2007), Art. ID rnm125, 48. MR 2377012, DOI 10.1093/imrn/rnm125
Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 13, 447–450 (French, with English summary). MR 660739
Alain Lascoux and Marcel-Paul Schützenberger, Keys & standard bases, Invariant theory and tableaux (Minneapolis, MN, 1988) IMA Vol. Math. Appl., vol. 19, Springer, New York, 1990, pp. 125–144. MR 1035493
I. G. Macdonald, Notes on Schubert polynomials, LACIM, Univ. Quebec à Montréal, Montréal, PQ, 1991.
I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Astérisque 237 (1996), Exp. No. 797, 4, 189–207. Séminaire Bourbaki, Vol. 1994/95. MR 1423624
S. Mason, A decomposition of Schur functions and an analogue of the Robinson-Schensted-Knuth algorithm, Sém. Lothar. Combin. 57 (2006/08), Art. B57e, 24. MR 2461993
S. Mason, An explicit construction of type A Demazure atoms, J. Algebraic Combin. 29 (2009), no. 3, 295–313. MR 2496309, DOI 10.1007/s10801-008-0133-4
Cara Monical, Set-Valued Skyline Fillings, preprint (2016), arXiv:1611.08777.
Cara Monical, Oliver Pechenik, and Dominic Searles, Families of polynomials from combinatorial ${K}$-theory, preprint, arXiv:1806.03802 (2018).
Eric M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), no. 1, 75–121. MR 1353018, DOI 10.1007/BF02392487
Oliver Pechenik and Dominic Searles, Decompositions of Grothendieck polynomials, Int. Math. Res. Not. IMRN 10 (2019), 3214–3241. MR 3952563, DOI 10.1093/imrn/rnx207
Anna Ying Pun, On decomposition of the product of Demazure atoms and Demazure characters, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–University of Pennsylvania. MR 3553609
Victor Reiner and Mark Shimozono, Key polynomials and a flagged Littlewood-Richardson rule, J. Combin. Theory Ser. A 70 (1995), no. 1, 107–143. MR 1324004, DOI 10.1016/0097-3165(95)90083-7
Vasu V. Tewari and Stephanie J. van Willigenburg, Modules of the 0-Hecke algebra and quasisymmetric Schur functions, Adv. Math. 285 (2015), 1025–1065. MR 3406520, DOI 10.1016/j.aim.2015.08.012