论文标题
舒伯特多项式的补充
Complements of Schubert polynomials
论文作者
论文摘要
令$ \ mathfrak {s} _W（x）$为$ \ {1,2，\ ldots，n \} $的置换$ w $的schubert polyenmial。对于任何给定的成分$μ$，我们说$ x^μ\ mathfrak {s} _W（x^{ - 1}）$是$ \ mathfrak {s} _w（x）$相对于$μ$的补充。当$μ$的每个部分等于$ n-1 $，嗯，玛莎恩，梅萨罗斯和st。他们进一步猜想$ \ mathfrak {s} _W（x）$的归一化是Lorentzian。可以证明，如果存在成分$μ$，这样$ x^μ\ mathfrak {s} _W（x^{ - 1}）$是schubert polyenmial，那么$ \ m m mathfrak {s} _W（x）$的归一化将是lorentzian。这激发了我们调查$ x^μ\ mathfrak {s} _W（x^{ - 1}）$的问题。我们表明，如果$ x^μ\ mathfrak {s} _W（x^{ - 1}）$是舒伯特多项式，那么$μ$必须是分区。我们还考虑$μ$是楼梯分区$Δ_n=（n-1，\ ldots，1,0）$，并获得该$ x^{δ_n} \ Mathfrak {s} _W（x^{ - 1}）$是schubert polynomial if且仅在$ w $ w的情况下以$ w的范围来避免$ 132和3122。 $ x^μ\ mathfrak {s} _W（x^{ - 1}）$是Schubert多项式。
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.
