偏序集的星状几何
Stellahedral geometry of partially ordered sets
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中文总结 AI 辅助
本文提出偏序集的星状变换，保持欧拉性等性质，并证明其与环面h-多项式联系，从而得出增广Chow多项式的正性结果，并否定两个开放问题。
中文摘要 AI 辅助
我们引入了一种偏序集上的变换，称为星状变换，它具有显著的性质。它保持欧拉性、Cohen-Macaulay 性以及作为多胞形的面偏序集的性质。此外，对于凸多胞形，它允许一个显式的几何实现，并特化为将单纯形映射到星状体的构造。该定义的一个动机来自环面 $h$-多项式和欧拉偏序集的（增广）Chow 多项式的理论。我们证明了欧拉偏序集 $P$ 的右增广 Chow 多项式与 $P$ 的星状变换的环面 $h$-多项式一致。我们利用这一视角，结合 Ehrenborg (2005) 和 Karu (2006) 的 $\mathbf{cd}$-指数结果，证明了增广 Chow 多项式的两个正性结果：对于 Gorenstein* 偏序集，它们是单峰的；对于多胞形的面偏序集，它们是 $\gamma$-正的。在此过程中，我们对关于欧拉和 Gorenstein* 偏序集的两个开放问题给出了否定答案。第一个是 Ferroni、Matherne 和 Vecchi (2024) 提出的关于欧拉 Chow 多项式非负性的问题。第二个是 Athanasiadis 和 Kalampogia-Evangelinou (2023) 提出的关于 Gorenstein* 偏序集的链和 Chow 多项式实根性的问题：这些例子提供了 Murai 和 Nevo (2014) 引入的一种技术的新应用。
英文摘要
We introduce a transformation on partially ordered sets, termed the \emph{stellahedral transform}, with notable features. It preserves the properties of being Eulerian, Cohen--Macaulay, and of being the face poset of a polytope. Furthermore, it admits an explicit geometric realization for convex polytopes and specializes to the construction that takes a simplex to the stellahedron. One motivation for this definition comes from the theory of toric $h$-polynomials and (augmented) Chow polynomials of Eulerian posets. We show that the right augmented Chow polynomial of an Eulerian poset $P$ agrees with the toric $h$-polynomial of the stellahedral transform of $P$. We use this perspective, together with $\mathbf{cd}$-index results due to Ehrenborg (2005) and Karu (2006), to prove two positivity results for augmented Chow polynomials: for Gorenstein* posets they are unimodal, and for face posets of polytopes they are $γ$-positive. Along the way we provide negative answers to two open questions concerning Eulerian and Gorenstein* posets. First, the question on the nonnegativity of Eulerian Chow polynomials, posed by Ferroni, Matherne, and Vecchi (2024). Second, the question posed by Athanasiadis and Kalampogia-Evangelinou (2023) on the real-rootedness of chain and Chow polynomials of Gorenstein* posets: these examples provide a novel application of a technique introduced by Murai and Nevo (2014).
发表机构
- University of Warwick（华威大学）
- Università di Pisa（比萨大学）
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