Combinatorial Structures in Cluster Algebras

簇变异是一定的高度对称和结构化的几何空间。在过去的二十年里，聚类变异在几何、表示理论和数学物理中都出现了。聚类代数是一种代数结构，它允许我们计算聚类变量并理解它们的结构。为了更好地了解集群品种，本研究项目进行了两条调查路线。该项目为本科生和研究生提供了培训机会。研究的第一步是构建辫状品种、理查德森品种及其相关空间的集群结构。这些几何对象最初出现在表征理论中，最近被发现在结理论中发挥重要作用。在这些空间上构建集群结构将允许我们使用集群理论中的所有技术来探索这些空间。研究的第二条线是发展集群品种和考克斯特群之间的关系，考克斯特群描述了反射镜集合的对称性。这已经被广泛地用于“有限型”的簇代数，它对应于具有有限多反射的Coxeter群，但类似的故事应该存在于所有的簇代数中，本项目建议发展它。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Cluster varieties are certain highly symmetric and structured geometric spaces. Over the last twenty years, cluster varieties have turned up throughout geometry, representation theory and mathematical physics. Cluster algebras are algebraic structures which allow us to compute with cluster varieties and understand their structure. This research project pursues two lines of investigation in order to better understand cluster varieties. The project provides training opportunities for both undergraduate and graduate students.The first line of investigation is to construct cluster structures on braid varieties, Richardson varieties, and related spaces. These are geometric objects which originally arose in representation theory and have recently been discovered to play important roles in knot theory. Constructing cluster structures on these spaces will allow us to use all the techniques from cluster theory to explore these spaces. The second line of investigation is to develop relationships between cluster varieties and Coxeter groups, which describe the symmetries of collections of reflecting mirrors. This has been extensively done for cluster algebras "of finite type", which correspond to Coxeter groups with finitely many reflections, but an analogous story should exist for all cluster algebras, and this project proposes to develop it.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.