ALGEBRAIC COMBINATORICS

no. 2

Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 301-307.

We study the action of a differential operator on Schubert polynomials. Using this action, we first give a short new proof of an identity of I. Macdonald (1991). We then prove a determinant conjecture of R. Stanley (2017). This conjecture implies the (strong) Sperner property for the weak order on the symmetric group, a property recently established by C. Gaetz and Y. Gao (2019).

Received: 2019-01-03
Revised: 2019-08-05
Accepted: 2019-08-08
Published online: 2020-04-01
DOI: https://doi.org/10.5802/alco.93
Classification: 05E05,  06A07,  15A15,  05E10
Keywords: Sperner property, weak order, Schubert polynomial, Macdonald identity 
@article{ALCO_2020__3_2_301_0,
     author = {Hamaker, Zachary and Pechenik, Oliver and Speyer, David E and Weigandt, Anna},
     title = {Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley},
     journal = {Algebraic Combinatorics},
     pages = {301--307},
     publisher = {MathOA foundation},
     volume = {3},
     number = {2},
     year = {2020},
     doi = {10.5802/alco.93},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2020__3_2_301_0/}
}
Hamaker, Zachary; Pechenik, Oliver; Speyer, David E; Weigandt, Anna. Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 301-307. doi : 10.5802/alco.93. https://alco.centre-mersenne.org/item/ALCO_2020__3_2_301_0/

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

[2] Billey, Sara C.; Holroyd, Alexander E.; Young, Benjamin J. A bijective proof of Macdonald’s reduced word formula, Algebraic Combin., Volume 2 (2019) no. 2, pp. 217-248 | Article | MR 3934829 | Zbl 1409.05024

[3] Fomin, Sergey; Stanley, Richard P. Schubert polynomials and the nil-Coxeter algebra, Adv. Math., Volume 103 (1994) no. 2, pp. 196-207 | Article | MR 1265793 | Zbl 0809.05091

[4] Gaetz, Christian; Gao, Yibo A combinatorial duality between the weak and strong Bruhat orders (2018), 14 pages (https://arxiv.org/abs/1812.05126) | Zbl 07172386

[5] Gaetz, Christian; Gao, Yibo A combinatorial 𝔰𝔩 2 -action and the Sperner property for the weak order (2019) (Proc. Amer. Math. Soc., to appear, https://arxiv.org/abs/1811.05501) | Zbl 07144479

[6] Macdonald, Ian Grant Notes on Schubert polynomials Volume 6, Publications du LACIM, Université du Québec à Montréal, 1991

[7] Manivel, Laurent Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, Volume 6, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001, viii+167 pages (Translated from the 1998 French original by John R. Swallow, Cours Spécialisés [Specialized Courses], 3) | MR 1852463 | Zbl 0998.14023

[8] Proctor, Robert A. Product evaluations of Lefschetz determinants for Grassmannians and of determinants of multinomial coefficients, J. Comb. Theory, Ser. A, Volume 54 (1990) no. 2, pp. 235-247 | Article | MR 1059998 | Zbl 0736.05004

[9] Stanley, Richard P. Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods, Volume 1 (1980) no. 2, pp. 168-184 | Article | MR 578321 | Zbl 0502.05004

[10] Stanley, Richard P. Some Schubert shenanigans (2017), 8 pages (https://arxiv.org/abs/1704.00851)