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

TL;DR

This paper proves that all 132-avoiding intervals in the weak order of the symmetric group have the Sperner property, extending previous results and providing new algebraic and combinatorial insights.

Contribution

It generalizes the Sperner property to all 132-avoiding intervals in the weak order, using an $rak{sl}_2$-action, and introduces a new formula for Schubert polynomial specializations.

Findings

Weak order on 132-avoiding intervals has the Sperner property.

Established an $rak{sl}_2$-action respecting the weak order.

Derived a new formula for principal specializations of Schubert polynomials.

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.