论文标题
多边形群集代数的膨胀posets
Expansion Posets for Polygon Cluster Algebras
论文作者
论文摘要
将膨胀poset定义为在多边形中附着在弧上的群集变量的单元的poset，其中每个单个单元都由某些固定组合群群膨胀公式由相应的组合对象表示。我们对与类型$ a $ a $表面群集代数相关的几个相互关联的组合对象和结构进行了介绍，包括某些类别的弧，三角形和分布晶格。我们使用这些互动来为ARC制定双重版本的双重关系，以及三个现有扩展Posets的双重版本。特别是，这导致了两个新的群集扩展公式，并恢复了Propp等人的晶格路径膨胀。我们从双弧线中提供了扩展poset及其双重版本之间的明确，具有结构的POSET同构。我们还表明，从同一弧构建的扩展Poset及其双重版本在分布晶格的意义上是双重的。 我们表明，任何扩展poset均与封闭的间隔同构，其中一个lattices $ l（m，n）$的年轻图中包含的$ m \ times n $ grid中包含的年轻图，任何$ l（m，n）$都具有此类间隔。我们给出了两个晶格路径扩展Poset的等级函数的两个公式，并证明每当基础蛇形由最多四个最大直段构建时，该等级函数都是单峰的。我们表明，任何类型的$ a $ cluster变量的支持是groupoid的轨道。最后，在与尼古拉斯烤箱的工作联合中，我们将$ t $ paths概括为仿射标志的配置，并证明当初始种子来自风扇三角形时，$ t $ path的扩展类似于$ $ a $ a $ a $ case。
Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some fixed combinatorial cluster expansion formula. We introduce an involution on several of the interrelated combinatorial objects and constructions associated to type $A$ surface cluster algebras, including certain classes of arcs, triangulations, and distributive lattices. We use these involutions to formulate a dual version of skein relations for arcs, and dual versions of three existing expansion posets. In particular, this leads to two new cluster expansion formulas, and recovers the lattice path expansion of Propp et al. We provide an explicit, structure-preserving poset isomorphism between an expansion poset and its dual version from the dual arc. We also show that an expansion poset and its dual version constructed from the same arc are dual in the sense of distributive lattices. We show that any expansion poset is isomorphic to a closed interval in one of the lattices $L(m,n)$ of Young diagrams contained in an $m \times n$ grid, and that any $L(m,n)$ has a covering by such intervals. We give two formulas for the rank function of any lattice path expansion poset, and prove that this rank function is unimodal whenever the underlying snake graph is built from at most four maximal straight segments. We show that the support of any type $A$ cluster variable is the orbit of a groupoid. Finally, in work joint with Nicholas Ovenhouse, we partially generalize $T$-paths to configurations of affine flags, and prove that a $T$-path expansion analogous to the type $A$ case holds when the initial seed is from a fan triangulation.
