Browse issues
no. 2
A Demazure crystal construction for Schubert polynomials
Assaf, Sami1; Schilling, Anne2
1 Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, U.S.A.
2 Department of Mathematics, UC Davis, One Shields Ave., Davis, CA 95616-8633, U.S.A.
Algebraic Combinatorics, Volume 1 (2018) no. 2, pp. 225-247.
corrected-by Corrigendum: A Demazure crystal construction for Schubert polynomials

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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.13
Classification: 14N15, 05E10, 05A05, 05E05, 05E18, 20G42
Mots-clés : Schubert polynomials, Demazure characters, Stanley symmetric functions, crystal bases

Assaf, Sami 1; Schilling, Anne 2

1 Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, U.S.A.
2 Department of Mathematics, UC Davis, One Shields Ave., Davis, CA 95616-8633, U.S.A.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2018__1_2_225_0,
     author = {Assaf, Sami and Schilling, Anne},
     title = {A {Demazure} crystal construction for {Schubert} polynomials},
     journal = {Algebraic Combinatorics},
     pages = {225--247},
     publisher = {MathOA foundation},
     volume = {1},
     number = {2},
     year = {2018},
     doi = {10.5802/alco.13},
     mrnumber = {3856523},
     zbl = {06882340},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.13/}
}
TY  - JOUR
AU  - Assaf, Sami
AU  - Schilling, Anne
TI  - A Demazure crystal construction for Schubert polynomials
JO  - Algebraic Combinatorics
PY  - 2018
SP  - 225
EP  - 247
VL  - 1
IS  - 2
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.13/
DO  - 10.5802/alco.13
LA  - en
ID  - ALCO_2018__1_2_225_0
ER  - 
%0 Journal Article
%A Assaf, Sami
%A Schilling, Anne
%T A Demazure crystal construction for Schubert polynomials
%J Algebraic Combinatorics
%D 2018
%P 225-247
%V 1
%N 2
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.13/
%R 10.5802/alco.13
%G en
%F ALCO_2018__1_2_225_0
Assaf, Sami; Schilling, Anne. A Demazure crystal construction for Schubert polynomials. Algebraic Combinatorics, Volume 1 (2018) no. 2, pp. 225-247. doi : 10.5802/alco.13. https://alco.centre-mersenne.org/articles/10.5802/alco.13/

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

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

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

[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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

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

[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 | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

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

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

[23] Monical, Cara Set-valued skyline fillings, Sém. Lothar. Combin., Volume 78B (2017), Paper no. Art. 35, 12 pages | MR | Zbl

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

Cited by Sources: