Cluster structures for positroid cells in the Grassmannian

这是一个代数项目，与组合学，李群理论和几何学有关。所提出的研究研究在多边形上的颤动。一个有向图是一个有许多顶点和箭头（有向边）的有向图。我们感兴趣的是把定向循环粘在一起而产生的特殊颤动。一个对偶概念是多边形中的链图：它是多边形中定向曲线的集合，这些曲线连接多边形的顶点，在每个顶点处，一条曲线开始，一条曲线结束。沿着多边形周围的曲线描述其顶点的排列。这两个对象已在描述集群代数和集群类别结构的坐标环的格拉斯曼各种k维子空间在n维复杂的空间，一个经典的对象在代数几何。这是第一个例子之一的协调环，这是一个集群代数：连接到年轻的主题集群代数和集群类别导致了大量的研究活动的格拉斯曼和相关的代数品种，使这一领域非常活跃。目的和目标：在项目中，我们将介绍对箭盒和图的操作，如切割顶点或曲线。这将改变置换，并预期导致格拉斯曼的更小维度的细胞（这些是更小维度的代数簇，可以被视为它的退化）。Grassmannian Gr（2，n）被很好地理解，并为此提供了一个很好的起点：目标是形式化从Gr（2，n）到Gr（k，n）的退化算法，其中k = 3，4等，从k = 3开始。该项目还将涉及研究相关的根系以及它们在操作下如何变化。在集群类别方面，确定相关的模块类别将是有趣的。该项目是跨学科的，在代数，组合学，李理论和几何的接口，导致在一套不同的方法。潜在的应用是一个组合的方法，一方面正向细胞的集群类别和偏序集描述的正向细胞所产生的退化算法。

This is a project in algebra, with links to combinatorics, Lie theory and geometry. The proposed research studies quivers on polygons. A quiver is an oriented graph with finitely many vertices and arrows (oriented edges) between them. We are interested in special quivers which arise from gluing oriented cycles together. A dual notion to these is a strand diagram in the polygon: it is a collection of oriented curves in the polygon, these curves link the vertices of the polygon, at each vertex one curve starts and one curve ends. Following the curves around the polygon describes a permutation of its vertices. Both these objects have been instrumental in describing cluster algebra and cluster category structure of the coordinate ring of the Grassmannian variety of k-dimensional subspaces in n-dimensional complex space, a classical object in algebraic geometry. This is among one of the first examples of a coordinate ring which is a cluster algebra: The connection to the young subject of cluster algebras and cluster categories has led to a lot of research activities on the Grassmannian and related algebraic varieties and makes this a very active field. Aims and objectives: In the project, we will introduce operations on the quivers and on the diagrams, like cutting vertices or curves. This will change the permutation and is expected to result in smaller dimensional cells of the Grassmannian (these are algebraic varieties of smaller dimension which can be viewed as degenerations of it). The Grassmannian Gr(2,n) is well understood and provides a good starting point for this: The goal is to formalize a degeneration algorithm from Gr(2,n) to Gr(k,n) for k=3,4, etc., starting with k=3. The project will also involve studying associated root systems and how they change under the operations. On the cluster category side, it will be interesting to determine associated module categories. The project is intradisciplinary, at the interface of algebra, combinatorics, Lie theory and geometry, resulting in a diverse set of methods. Potential applications are a combinatorial approach to the cluster categories for positroid cells on one hand and a poset description of positroid cells arising from the degeneration algorithm.