Complements of Schubert polynomials

Abstract

Let $\mathfrak{S}_w(x)$ be the Schubert polynomial for a permutation $w$ of $\{1,2,\ldots,n\}$. For any given composition $\mu$, we say that $x^\mu \mathfrak{S}_w(x^{-1})$ is the complement of $\mathfrak{S}_w(x)$ with respect to $\mu$. When each part of $\mu$ is equal to $n-1$, Huh, Matherne, M\'esz\'aros and St.\,Dizier proved that the normalization of $x^\mu \mathfrak{S}_w(x^{-1})$ is a Lorentzian polynomial. They further conjectured that the normalization of $\mathfrak{S}_w(x)$ is Lorentzian. It can be shown that if there exists a composition $\mu$ such that $x^\mu \mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then the normalization of $\mathfrak{S}_w(x)$ will be Lorentzian. This motivates us to investigate the problem of when $x^\mu \mathfrak{S}_w(x^{-1})$ is a Schubert polynomial. We show that if $x^\mu \mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then $\mu$ must be a partition. We also consider the case when $\mu$ is the staircase partition $\delta_n=(n-1,\ldots, 1,0)$, and obtain that $x^{\delta_n} \mathfrak{S}_w(x^{-1})$ is a Schubert polynomial if and only if $w$ avoids the patterns 132 and 312. A conjectured characterization of when $x^\mu \mathfrak{S}_w(x^{-1})$ is a Schubert polynomial is proposed.