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 $μ$, we say that $x^μ\mathfrak{S}_w(x^{-1})$ is the complement of $\mathfrak{S}_w(x)$ 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 $x^μ\mathfrak{S}_w(x^{-1})$ 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 $μ$ such that $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then the normalization of $\mathfrak{S}_w(x)$ will be Lorentzian. This motivates us to investigate the problem of when $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial. We show that if $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial, then $μ$ must be a partition. We also consider the case when $μ$ is the staircase partition $δ_n=(n-1,\ldots, 1,0)$, and obtain that $x^{δ_n} \mathfrak{S}_w(x^{-1})$ is a Schubert polynomial if and only if $w$ avoids the patterns 132 and 312. A conjectured characterization of when $x^μ\mathfrak{S}_w(x^{-1})$ is a Schubert polynomial is proposed.

Figures (8)

• Figure 2.1: An RC-graph of $w=15342$.
• Figure 2.2: The bottom and top RC-graphs of $w=25143$.
• Figure 2.3: A ladder move.
• Figure 3.4: The matrix representation of $w=426315$.
• Figure 3.5: The matrix representation of $w$.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Proposition 2.1: Bergeron--Billey Ber
• Proposition 2.2: Bergeron--Billey Ber
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4