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no. 5
Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials
Assaf, Sami1; González, Nicolle2
1 Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, U.S.A.
2 UCLA Department of Mathematics, Los Angeles CA 90095-1555, U.S.A.
Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 777-793.

We give a crystal-theoretic proof that nonsymmetric Macdonald polynomials specialized to t=0 are affine Demazure characters. We explicitly construct an affine Demazure crystal on semistandard key tabloids such that removing the affine edges recovers the finite Demazure crystals constructed earlier by the authors. We also realize the filtration on highest weight modules by Demazure modules by defining explicit embedding operators which, at the level of characters, parallels the recursion operators of Knop and Sahi for specialized nonsymmetric Macdonald polynomials. Thus we prove combinatorially in type A that every affine Demazure module admits a finite Demazure flag.

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Accepted:
Published online:
DOI: 10.5802/alco.178
Classification: 05E10, 05E05, 05A05, 05E18, 17B37
Keywords: Affine Demazure crystal, affine Demazure character, nonsymmetric Macdonald polynomial

Assaf, Sami 1; González, Nicolle 2

1 Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, U.S.A.
2 UCLA Department of Mathematics, Los Angeles CA 90095-1555, U.S.A.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Assaf, Sami; González, Nicolle. Affine Demazure crystals for specialized nonsymmetric Macdonald polynomials. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 777-793. doi : 10.5802/alco.178. https://alco.centre-mersenne.org/articles/10.5802/alco.178/

[1] Andersen, Henning H. Schubert varieties and Demazure’s character formula, Invent. Math., Volume 79 (1985) no. 3, pp. 611-618 | DOI | MR | Zbl

[2] Assaf, Sami Weak dual equivalence for polynomials (https://arxiv.org/abs/1702.04051)

[3] Assaf, Sami Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials, Trans. Amer. Math. Soc., Volume 370 (2018) no. 12, pp. 8777-8796 | DOI | MR | Zbl

[4] Assaf, Sami; González, Nicolle Demazure crystals for specialized nonsymmetric Macdonald polynomials, J. Combin. Theory Ser. A, Volume 182 (2021), Paper no. 105463 | DOI | MR

[5] Assaf, Sami; Schilling, Anne A Demazure crystal construction for Schubert polynomials, Algebr. Comb., Volume 1 (2018) no. 2, pp. 225-247 | DOI | MR | Zbl

[6] Butler, Lynne M. Combinatorial properties of partially ordered sets associated with partitions and finite abelian groups, Ph. D. Thesis, Massachusetts Institute of Technology (1986)

[7] Butler, Lynne M. Subgroup lattices and symmetric functions, Mem. Amer. Math. Soc., Volume 112 (1994) no. 539, p. vi+160 | DOI | MR | Zbl

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

[9] Demazure, Michel Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4), Volume 7 (1974), pp. 53-88 | MR | Zbl

[10] Demazure, Michel Une nouvelle formule des caractères, Bull. Sci. Math. (2), Volume 98 (1974) no. 3, pp. 163-172 | MR | Zbl

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

[12] Ion, Bogdan Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J., Volume 116 (2003) no. 2, pp. 299-318 | DOI | MR | Zbl

[13] Kashiwara, Masaki On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J., Volume 63 (1991) no. 2, pp. 465-516 | DOI | MR | Zbl

[14] Kashiwara, Masaki The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., Volume 71 (1993) no. 3, pp. 839-858 | DOI | MR | Zbl

[15] Kashiwara, Masaki On level-zero representations of quantized affine algebras, Duke Math. J., Volume 112 (2002) no. 1, pp. 117-175 | DOI | MR | Zbl

[16] Kashiwara, Masaki; Nakashima, Toshiki Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra, Volume 165 (1994) no. 2, pp. 295-345 | DOI | MR | Zbl

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

[18] Kumar, Shrawan Demazure character formula in arbitrary Kac–Moody setting, Invent. Math., Volume 89 (1987) no. 2, pp. 395-423 | DOI | MR | Zbl

[19] Lascoux, Alain; Schützenberger, Marcel-Paul Sur une conjecture de H. O. Foulkes, C. R. Acad. Sci. Paris Sér. A-B, Volume 286 (1978) no. 7, p. A323-A324

[20] Littelmann, Peter Crystal graphs and Young tableaux, J. Algebra, Volume 175 (1995) no. 1, pp. 65-87 | DOI | MR | Zbl

[21] Macdonald, Ian G. A new class of symmetric functions, Actes du 20e Seminaire Lotharingien, Volume 372 (1988), pp. 131-171

[22] Macdonald, Ian 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

[23] MacDonald, Ian G. Affine Hecke algebras and orthogonal polynomials, Séminaire Bourbaki : volume 1994/95, exposés 790-804 (Astérisque), Société mathématique de France, 1996 no. 237 (talk:797) | MR | Zbl

[24] Nakayashiki, Atsushi; Yamada, Yasuhiko Kostka polynomials and energy functions in solvable lattice models, Selecta Math. (N.S.), Volume 3 (1997) no. 4, pp. 547-599 | DOI | MR | Zbl

[25] Opdam, Eric M. Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., Volume 175 (1995) no. 1, pp. 75-121 | DOI | MR | Zbl

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

[27] Sanderson, Yasmine B. On the connection between Macdonald polynomials and Demazure characters, J. Algebraic Combin., Volume 11 (2000) no. 3, pp. 269-275 | DOI | MR | Zbl

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