流多面体的三角剖分、充足标架与Gentle代数
Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras
AI总结：
该研究刻画了容许充足标架的有向无环图特征并枚举其数量，建立了DKK三角剖分极大单纯形与gentle代数τ-倾斜偏序集的联系，证明完全DAG的流多面体为Gorenstein且具有单峰Ehrhart h*-多项式。
摘要：
有向无环图（DAG）的非负流锥可由DAG的标架诱导出正则幺模三角剖分。这些三角剖分可限制为强度为1的流对应的流多面体的三角剖分，称为DKK三角剖分。对于一类被称为充足标架的特殊标架，流锥的这些三角剖分可投影为完全扇。我们刻画了容许充足标架的DAG的特征，并枚举了固定DAG的充足标架数量。我们建立了DKK三角剖分中的极大单纯形与某类gentle代数的τ-倾斜偏序集之间的联系，这使得我们能为带充足标架的DAG的任意DKK三角剖分的对偶图赋予偏序结构。利用这一联系，我们证明了对于完全DAG（即内部顶点的入度和出度均为2的DAG），其流多面体是Gorenstein的，且具有单峰的Ehrhart h*-多项式。
英文摘要：
The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope for strength one flows, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. We characterize the DAGs that admit ample framings, and we enumerate the number of ample framings for a fixed DAG. We establish a connection between maximal simplices in DKK triangulations and $τ$-tilting posets for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that for full DAGs, i.e., those DAGs with inner vertices having in-degree and out-degree equal to two, the flow polytopes are Gorenstein and have unimodal Ehrhart $h^\ast$-polynomials.
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