有向翻转图与不交叉树划分
Oriented Flip Graphs and Noncrossing Tree Partitions
摘要：
本文研究了Coelho Simoes和Parsons引入的一类表示有限的温柔代数（称为tiling代数）的模范畴中扭对偏序集的格性质.我们提出了一个利用凸多边形的多边形细分的扭对组合模型。我们使用这个模型和格理论分类2-term简单的集合在相应的平铺代数的有界导出范畴。作为结果，我们得到的c-矩阵的一个特征，任何的突变等价于A型Dynkin矩阵。我们的模型是使用多边形细分的对偶树。给定这样一棵树，我们引入一个由树支撑的非交叉测地线的单纯复形，我们称之为非交叉复形。非交叉复形的刻面可以被赋予有向翻转图的结构。可以用这种方式表示的定向翻转图的特殊情况包括Tamari阶、A型寒武纪阶、箭图突变的定向交换图-等价于路径交换。我们证明了任何非交叉复形的有向翻转图是一个多边形的同余一致格。为了做到这一点，我们表示定向翻转图作为双闭集的格的格商。非交叉复合体的小平面有一个交替的顺序，称为碎片交叉顺序。我们证明，这个碎片的交叉顺序是同构的非交叉树分区的格。有向翻转图从它的同余一致结构继承了循环作用。在非交叉树分区，这种循环行动推广了经典的Kreweras互补的非交叉集分区。我们表明，数据的非交叉树分区及其Kreweras补相当于一个2项简单的头脑收集相关的平铺代数。
英文摘要：
In this paper, we study the lattice properties of posets of torsion pairs in the module category of a family of representation-finite gentle algebras called tiling algebras, introduced by Coelho Simoes and Parsons. We present a combinatorial model for torsion pairs using polyogonal subdivisions of a convex polygon. We use this model and the lattice theory to classify 2-term simple-minded collections in the bounded derived category of the corresponding tiling algebra. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver. Our model is developed using the dual tree of a polygonal subdivision. Given such a tree, we introduce a simplicial complex of noncrossing geodesics supported by the tree which we call the noncrossing complex. The facets of the noncrossing complex may be given the structure of an oriented flip graph. Special cases of the oriented flip graphs that may be expressed in this way include the Tamari order, type A Cambrian orders, oriented exchange graphs for quivers mutation-equivalent to a path quiver. We prove that the oriented flip graph of any noncrossing complex is a polygonal, congruence-uniform lattice. To do so, we express the oriented flip graph as a lattice quotient of a lattice of biclosed sets. The facets of the noncrossing complex have an alternate ordering known as the shard intersection order. We prove that this shard intersection order is isomorphic to a lattice of noncrossing tree partitions. The oriented flip graph inherits a cyclic action from its congruence-uniform structure. On noncrossing tree partitions, this cyclic action generalizes the classical Kreweras complementation on noncrossing set partitions. We show that the data of a noncrossing tree partition and its Kreweras complement is equivalent to a 2-term simple-minded collection of the associated tiling algebra.
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