Cluster structures for positroid cells in the Grassmannian

这是一个代数项目，与组合学、李理论和几何学有联系。拟议的研究针对的是多边形上的抖动。箭图是一个有向图，它们之间有有限多的顶点和箭头(有向边)。我们对粘合定向循环所产生的特殊颤动感兴趣。其中的双重概念是多边形中的链状图：它是多边形中定向曲线的集合，这些曲线链接多边形的顶点，在每个顶点处，一条曲线开始，一条曲线结束。跟随多边形周围的曲线描述其顶点的排列。这两个对象都有助于描述n维复空间中k维子空间的Grassmanian簇的坐标环的簇代数和簇范畴结构，n维复空间是代数几何中的经典对象。这是作为簇代数的坐标环的首批例子之一：与簇代数和簇范畴这一年轻学科的联系导致了许多关于Grassman代数及其相关代数簇的研究活动，并使这一领域成为一个非常活跃的领域。目标和目标：在这个项目中，我们将介绍对箭图和图表的操作，如切割顶点或曲线。这将改变排列，并有望导致Grassmanian的较小维度的单元(这些是较小维度的代数变体，可以被视为它的退化)。GrassmannianGr(2，n)是一个很好的起点：目标是形式化一个从Gr(2，n)到Gr(k，n)(k=3，4等)的退化算法，从k=3开始。该项目还将研究相关的根系统以及它们在运算中如何变化。在集群类别方面，确定相关的模块类别将是很有趣的。该项目是跨学科的，涉及代数、组合学、测谎论和几何学，产生了一套不同的方法。潜在的应用是一种组合方法，一方面是对正态细胞的簇类别的组合方法，另一方面是对由退化算法产生的正态细胞的偏序集描述。

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.