Upper Bounds of Schubert Polynomials

TL;DR

This paper characterizes when Schubert and key polynomials reach their upper bounds based on permutation and composition pattern avoidance, revealing new combinatorial and algebraic properties.

Contribution

It provides a complete pattern avoidance characterization for the upper bounds of Schubert and key polynomials, linking combinatorics with algebraic geometry.

Findings

Schubert polynomial reaches upper bound iff permutation avoids patterns 1432 and 1423.

Key polynomial reaches upper bound iff composition avoids pattern (0,2).

When avoiding (0,2), key polynomials are Lorentzian, supporting a conjecture.

Abstract

Let $w$ be a permutation of $\{1,2,\ldots,n \}$, and let $D(w)$ be the Rothe diagram of $w$. The Schubert polynomial $\mathfrak{S}_w(x)$ can be realized as the dual character of the flagged Weyl module associated to $D(w)$. This implies a coefficient-wise inequality \[\mathrm{Min}_w(x)\leq \mathfrak{S}_w(x)\leq \mathrm{Max}_w(x),\] where both $\mathrm{Min}_w(x)$ and $\mathrm{Max}_w(x)$ are polynomials determined by $D(w)$. Fink, M\'esz\'aros and St.$\,$Dizier found that $\mathfrak{S}_w(x)$ equals the lower bound $\mathrm{Min}_w(x)$ if and only if $w$ avoids twelve permutation patterns. In this paper, we show that $\mathfrak{S}_w(x)$ reaches the upper bound $\mathrm{Max}_w(x)$ if and only if $w$ avoids two permutation patterns 1432 and 1423. Similarly, for any given composition $\alpha\in \mathbb{Z}_{\geq 0}^n$, one can define a lower bound $\mathrm{Min}_\alpha(x)$ and an upper bound $\mathrm{Max}_\alpha(x)$ for the key polynomial $\kappa_\alpha(x)$. Hodges and Yong established that $\kappa_{\alpha}(x)$ equals $\mathrm{Min}_\alpha(x)$ if and only if $\alpha$ avoids five composition patterns. We show that $\kappa_{\alpha}(x)$ equals $\mathrm{Max}_\alpha(x)$ if and only if $\alpha$ avoids a single composition pattern $(0,2)$. As an application, we obtain that when $\alpha$ avoids $(0,2)$, the key polynomial $\kappa_{\alpha}(x)$ is Lorentzian, partially verifying a conjecture of Huh, Matherne, M\'esz\'aros and St.$\,$Dizier.