Athanasiadis-Chapoton 序数和猜想的一个证明
A proof of the Athanasiadis-Chapoton ordinal-sum conjecture
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中文总结 AI 辅助
本文证明了 Athanasiadis-Chapoton 关于有限预序的极预序多胞形 $h^*$-多项式在序数和下取乘积的猜想，通过构造分片幺模剪切给出格点双射，并推导出归一化体积的乘积公式。
中文摘要 AI 辅助
给定一个有限预序 $\tau$，我们研究其序关系如何决定关联的极预序多胞形的 $h^*$-多项式。设 $\mathcal R_{\tau}^{\vee}$ 为负标准基向量与非空序理想的指示向量的凸包，并记 $h_\tau^*(t)=h^*(\mathcal R_{\tau}^{\vee},t)$。Athanasiadis 和 Chapoton 猜想这一枚举不变量将序数和映射为乘积：$$h_{\tau_1\oplus\tau_2}^*(t)=h_{\tau_1}^*(t)\\,h_{\tau_2}^*(t)。$$ 我们证明该猜想对任意有限预序成立，包括具有非平凡等价类的预序。更精确地，我们构造一个从自由和 $\mathcal R_{\tau_1}^{\vee}\oplus\mathcal R_{\tau_2}^{\vee}$ 到 $\mathcal R_{\tau_1\oplus\tau_2}^{\vee}$ 的正齐次分片幺模剪切，从而在每个非负实数膨胀尺度下给出格点双射。关键在于用最小非负保序反向控制函数表示 Minkowski 泛函的公式：剪切后，该泛函变为两个因子泛函之和。结合自由和的 Ehrhart 乘积公式，这证明了该猜想以及相应的归一化体积乘积公式。
英文摘要
Given a finite preorder $τ$, we study how its order relation determines the $h^*$-polynomial of the associated polar preorder polytope. Let $\mathcal R_τ^{\vee}$ be the convex hull of the negative standard basis vectors and the indicator vectors of nonempty order ideals, and write $h_τ^*(t)=h^*(\mathcal R_τ^{\vee},t)$. Athanasiadis and Chapoton conjectured that this enumerative invariant takes ordinal sums to products: $$h_{τ_1\oplusτ_2}^*(t)=h_{τ_1}^*(t)\,h_{τ_2}^*(t).$$ We prove the conjecture for arbitrary finite preorders, including those with nontrivial equivalence classes. More precisely, we construct a positively homogeneous piecewise unimodular shear from the free sum $\mathcal R_{τ_1}^{\vee}\oplus\mathcal R_{τ_2}^{\vee}$ onto $\mathcal R_{τ_1\oplusτ_2}^{\vee}$, giving lattice-point bijections at every nonnegative real dilation scale. The key is a formula for the Minkowski functional in terms of the least nonnegative order-reversing majorant: after the shear, the functional becomes the sum of the two factor functionals. Together with the Ehrhart product formula for free sums, this proves the conjecture and the corresponding product formula for normalized volume.
发表机构
- Fudan University（复旦大学）
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