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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.93
Classification: 05E05, 06A07, 15A15, 05E10
Keywords: Sperner property, weak order, Schubert polynomial, Macdonald identity
Hamaker, Zachary 1; Pechenik, Oliver 2; Speyer, David E 2; Weigandt, Anna 2

1 Department of Mathematics University of Florida Gainesville, FL 32601, USA
2 Department of Mathematics University of Michigan Ann Arbor, MI 48109, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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/articles/10.5802/alco.93/}
}
TY  - JOUR
AU  - Hamaker, Zachary
AU  - Pechenik, Oliver
AU  - Speyer, David E
AU  - Weigandt, Anna
TI  - Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley
JO  - Algebraic Combinatorics
PY  - 2020
SP  - 301
EP  - 307
VL  - 3
IS  - 2
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.93/
DO  - 10.5802/alco.93
LA  - en
ID  - ALCO_2020__3_2_301_0
ER  - 
%0 Journal Article
%A Hamaker, Zachary
%A Pechenik, Oliver
%A Speyer, David E
%A Weigandt, Anna
%T Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley
%J Algebraic Combinatorics
%D 2020
%P 301-307
%V 3
%N 2
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.93/
%R 10.5802/alco.93
%G en
%F 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/articles/10.5802/alco.93/

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

[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 | DOI | MR | Zbl

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

[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

[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

[6] Macdonald, Ian Grant Notes on Schubert polynomials, 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, 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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

Cited by Sources: