The Sperner property for $132$-avoiding intervals in the weak order

Abstract

A well-known result of Stanley from 1980 implies that the weak order on a maximal parabolic quotient of the symmetric group $S_n$ has the Sperner property; this same property was recently established for the weak order on all of $S_n$ by Gaetz and Gao, resolving a long-open problem. In this paper we interpolate between these results by showing that the weak order on any parabolic quotient of $S_n$ (and more generally on any $132$-avoiding interval) has the Sperner property. This result is proven by exhibiting an action of ${\mathfrak{sl}}_2$ respecting the weak order on these intervals. As a corollary we obtain a new formula for principal specializations of Schubert polynomials. Our formula can be seen as a strong Bruhat order analogue of Macdonald's reduced word formula. This proof technique and formula generalize work of Hamaker, Pechenik, Speyer, and Weigandt and Gaetz and Gao.

18 pages. v2: fixed typos

References in corpus (3)

• Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley
• A combinatorial ${\mathfrak{sl}}_2$-action and the Sperner property for the weak order
• A combinatorial duality between the weak and strong Bruhat orders