Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley

Abstract

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 (2018).

6 pages

References in corpus (2)

• Some Schubert shenanigans
• A combinatorial ${\mathfrak{sl}}_2$-action and the Sperner property for the weak order

Cited by in corpus (6)

• The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
• Gröbner geometry of Schubert polynomials through ice
• A combinatorial duality between the weak and strong Bruhat orders
• Padded Schubert polynomials and weighted enumeration of Bruhat chains
• Differential operators on Schur and Schubert polynomials
• The Sperner property for $132$-avoiding intervals in the weak order