Complements of Schubert polynomials

Abstract

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