论文标题
两个Poset多型是突变等效的
Two poset polytopes are mutation-equivalent
论文作者
论文摘要
在[1]中的fano歧管的背景下，引入了晶格多层$ p $的组合突变$ \ mathrm {mutrm {mut} _w（p，f）$。在[1]中也证明了对于晶格polytope $ p \ subseteq n_ \ mathbb {r} $，其中包含其内部的起源，极性duals $ p^* \ subseteq m_ \ subseteq m_ \ mathbb {r} $ and $ \ \ \ \ \ \ \ m {mathrm {mut {mut {mut {mut} $ supseteq p，p，p，f）同一Ehrhart系列。为了扩展此框架，在本文中，我们介绍了Minkowski的合理多型和理性多面体尖锥的组合突变，以$ N_ \ Mathbb {R} $。我们还可以在双侧介绍组合突变$ m_ \ mathbb {r} $，我们可以在$ m_ \ mathbb {r} $中申请包含原点的每个理性polytope（不一定在内部）。作为组合突变扩展的应用，我们证明可以通过$ M_ \ Mathbb {r} $从$π$的$ M_ \ Mathbb {r} $中的组合突变序列获得Poset $π$的链多层。也就是说，同一poset $π$的阶层和链多层是突变等效的。
The combinatorial mutation $\mathrm{mut}_w(P,F)$ for a lattice polytope $P$ was introduced in the context of mirror symmetry for Fano manifolds in [1]. It was also proved in [1] that for a lattice polytope $P \subseteq N_\mathbb{R}$ containing the origin in its interior, the polar duals $P^* \subseteq M_\mathbb{R}$ and $\mathrm{mut}_w(P,F)^* \subseteq M_\mathbb{R}$ have the same Ehrhart series. For extending this framework, in this paper, we introduce the combinatorial mutation for the Minkowski sum of rational polytopes and rational polyhedral pointed cones in $N_\mathbb{R}$. We can also introduce the combinatorial mutation in the dual side $M_\mathbb{R}$, which we can apply for every rational polytope in $M_\mathbb{R}$ containing the origin (not necessarily in the interior). As an application of this extension of the combinatorial mutation, we prove that the chain polytope of a poset $Π$ can be obtained by a sequence of the combinatorial mutation in $M_\mathbb{R}$ from the order polytope of $Π$. Namely, the order polytope and the chain polytope of the same poset $Π$ are mutation-equivalent.
