Cluster structures for positroid cells in the Grassmannian

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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.