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).
Revised: 2019-08-05
Accepted: 2019-08-08
Published online: 2020-04-01
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/
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