A Demazure crystal construction for Schubert polynomials
Algebraic Combinatorics, Volume 1 (2018) no. 2, p. 225-247
Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by Morse and the second author in connection with Stanley symmetric functions.
Received : 2017-08-11
Revised : 2017-11-21
Accepted : 2017-12-27
Published online : 2018-03-02
DOI : https://doi.org/10.5802/alco.13
Classification:  14N15,  05E10,  05A05,  05E05,  05E18,  20G42
Keywords: Schubert polynomials, Demazure characters, Stanley symmetric functions, crystal bases
@article{ALCO_2018__1_2_225_0,
     author = {Assaf, Sami and Schilling, Anne},
     title = {A Demazure crystal construction for Schubert polynomials},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {1},
     number = {2},
     year = {2018},
     pages = {225-247},
     doi = {10.5802/alco.13},
     zbl = {06882340},
     language = {en},
     url = {http://alco.centre-mersenne.org/item/ALCO_2018__1_2_225_0}
}
Assaf, Sami;Schilling, Anne. A Demazure crystal construction for Schubert polynomials. Algebraic Combinatorics, Volume 1 (2018) no. 2, p. 225-247. doi : 10.5802/alco.13. https://alco.centre-mersenne.org/item/ALCO_2018__1_2_225_0/

[1] Assaf, Sami Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials (to appear in Trans. Amer. Math. Soc. )

[2] Assaf, Sami Combinatorial models for Schubert polynomials (2017) (https://arxiv.org/abs/1703.00088) | Zbl 1356.14039

[3] Assaf, Sami Weak dual equivalence for polynomials (2017) (https://arxiv.org/abs/1702.04051) | Zbl 1356.14039

[4] Assaf, Sami; Searles, Dominic Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams, Adv. in Math., Volume 306 (2017), pp. 89-122 | Article | MR 3581299 | Zbl 1356.14039

[5] Bergeron, Nantel; Billey, Sara RC-graphs and Schubert polynomials, Experiment. Math., Volume 2 (1993) no. 4, pp. 257-269 | Article | MR 1281474 | Zbl 0803.05054

[6] Bernstein, I. N.; Gel’Fand, I. M.; Gel’Fand, S. I. Schubert cells, and the cohomology of the spaces G/P, Uspehi Mat. Nauk, Volume 28 (1973) no. 3, pp. 1-26 | MR 429933 | Zbl 0289.57024

[7] Billey, Sara; Jockusch, William; Stanley, Richard P. Some combinatorial properties of Schubert polynomials, J. Algebraic Combin., Volume 2 (1993) no. 4, pp. 345-374 | Article | MR 1241505 | Zbl 0790.05093

[8] Bump, Daniel; Schilling, Anne Crystal bases. Representations and combinatorics, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2017), xii+279 pages | Article | MR 3642318 | Zbl 06690908

[9] Demazure, Michel Une nouvelle formule des caractères, Bull. Sci. Math. (2), Volume 98 (1974) no. 3, pp. 163-172 | MR 430001 | Zbl 0365.17005

[10] Edelman, Paul; Greene, Curtis Balanced tableaux, Adv. in Math., Volume 63 (1987) no. 1, pp. 42-99 | Article | MR 871081 | Zbl 0616.05005

[11] Hong, Jin; Kang, Seok-Jin Introduction to quantum groups and crystal bases, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Volume 42 (2002), xviii+307 pages | MR 1881971 | Zbl 1134.17007

[12] Kashiwara, Masaki The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J., Volume 71 (1993) no. 3, pp. 839-858 | MR 1240605 | Zbl 0794.17008

[13] 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 | Article | MR 1273277 | Zbl 0808.17005

[14] Kohnert, Axel Weintrauben, Polynome, Tableaux, Bayreuth. Math. Schr. (1991) no. 38, pp. 1-97 (Dissertation, Universität Bayreuth, Bayreuth, 1990) | MR 1132534 | Zbl 0755.05095

[15] Lascoux, Alain; Schützenberger, Marcel-Paul Polynômes de Schubert, C. R. Acad. Sci. Paris Sér. I Math., Volume 294 (1982) no. 13, pp. 447-450 | MR 660739 | Zbl 0495.14031

[16] Lascoux, Alain; Schützenberger, Marcel-Paul Schubert polynomials and the Littlewood-Richardson rule, Lett. Math. Phys., Volume 10 (1985) no. 2-3, pp. 111-124 | Article | MR 815233 | Zbl 0586.20007

[17] Lascoux, Alain; Schützenberger, Marcel-Paul Keys & standard bases, Invariant theory and tableaux (Minneapolis, MN, 1988), Springer, New York (IMA Vol. Math. Appl.) Volume 19 (1990), pp. 125-144 | MR 1035493 | Zbl 0815.20013

[18] Lenart, Cristian A unified approach to combinatorial formulas for Schubert polynomials, J. Algebraic Combin., Volume 20 (2004) no. 3, pp. 263-299 | Article | MR 2106961 | Zbl 1056.05146

[19] Littelmann, Peter Crystal graphs and Young tableaux, J. Algebra, Volume 175 (1995) no. 1, pp. 65-87 | Article | MR 1338967 | Zbl 0831.17004

[20] Macdonald, I. G. Notes on Schubert polynomials, LACIM, Univ. Quebec a Montreal, Montreal, PQ (1991)

[21] Macdonald, I. G. Schubert polynomials, Surveys in combinatorics, 1991 (Guildford, 1991), Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Volume 166 (1991), pp. 73-99 | Article | MR 1161461 | Zbl 0784.05061

[22] Mason, Sarah An explicit construction of type A Demazure atoms, J. Algebraic Combin., Volume 29 (2009) no. 3, pp. 295-313 | Article | MR 2496309 | Zbl 1210.05175

[23] Monical, Cara Set-valued skyline fillings, Sém. Lothar. Combin., Volume 78B (2017), 12 pages | MR 3678617 | Zbl 1384.05160

[24] Morse, Jennifer; Schilling, Anne Crystal approach to affine Schubert calculus, Int. Math. Res. Not. (2016) no. 8, pp. 2239-2294 | Article | MR 3519114

[25] Reiner, Victor; Shimozono, Mark Key polynomials and a flagged Littlewood-Richardson rule, J. Combin. Theory Ser. A, Volume 70 (1995) no. 1, pp. 107-143 | Article | MR 1324004 | Zbl 0819.05058

[26] Reiner, Victor; Shimozono, Mark Plactification, J. Algebraic Combin., Volume 4 (1995) no. 4, pp. 331-351 | Article | MR 1346889 | Zbl 0922.05049

[27] Stanley, Richard P. On the number of reduced decompositions of elements of Coxeter groups, European J. Combin., Volume 5 (1984) no. 4, pp. 359-372 | Article | MR 782057 | Zbl 0587.20002

[28] Stembridge, John R. A local characterization of simply-laced crystals, Trans. Amer. Math. Soc., Volume 355 (2003) no. 12, pp. 4807-4823 | Article | MR 1997585 | Zbl 1047.17007