Cluster structures for positroid cells in the Grassmannian

格拉斯曼阶正样细胞的簇结构

• 批准号: 2744564
• 金额: --
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2744564
• 关键词: Cluster; structures; positroid; cells; Grassmannian

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.

这是一个代数项目，与组合学、李理论和几何相关。拟议的研究围绕多边形进行。箭袋是一个有向图，具有有限多个顶点和它们之间的箭头（有向边）。我们对由定向循环粘合在一起而产生的特殊箭袋感兴趣。这些的双重概念是多边形中的链图：它是多边形中定向曲线的集合，这些曲线链接多边形的顶点，在每个顶点，一条曲线开始，一条曲线结束。沿着多边形周围的曲线描述了其顶点的排列。这两个对象都有助于描述 n 维复空间中 k 维子空间的格拉斯曼簇的坐标环的簇代数和簇类别结构，这是代数几何中的经典对象。这是作为簇代数的坐标环的第一个例子之一：与簇代数和簇范畴这一年轻学科的联系导致了对格拉斯曼和相关代数簇的大量研究活动，并使之成为一个非常重要的研究活动。活跃领域。目的和目标：在该项目中，我们将介绍对箭袋和图表的操作，例如切割顶点或曲线。这将改变排列，并预计会导致格拉斯曼元胞的尺寸更小（这些是更小尺寸的代数簇，可以被视为它的退化）。格拉斯曼 Gr(2,n) 很好理解，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（k=3,4 等） .，从 k=3 开始。该项目还将涉及研究相关的根系统以及它们在操作下如何变化。在集群类别方面，确定关联的模块类别将很有趣。该项目是跨学科的，处于代数、组合学、李理论和几何的交汇处，产生了一套多样化的方法。潜在的应用一方面是对正细胞簇类别的组合方法，另一方面是由退化算法产生的正细胞的偏序集描述。