Cluster structures on subvarieties of the Grassmannian

Abstract: Early in the history of cluster algebras, Scott showed that the homogeneous coordinate ring of the Grassmannian is a cluster algebra, with seeds given by Postnikov's plabic graphs for the Grassmannian. Recently the analogous statement has been proved for open Schubert varieties (Leclerc, Serhiyenko-SB-Williams) and more generally, for open positroid varieties (Galashin-Lam). I'll discuss joint work with Chris Fraser, in which we provide a family of cluster structures for each open positroid variety. Seeds for these cluster structures are given by relabeled plabic graphs, a natural generalization of Postnikov's construction. I'll also explain how for Schubert varieties (and conjecturally in general), relabeled plabic graphs give additional seeds for the standard" cluster structure. Towards the end, I'll also discuss joint work with M. Parisi and L. Williams on the cluster structure of some subvarieties of Gr(2, n) which arise naturally in the study of the m=2 amplituhedron. These subvarieties are closely related to positroid varieties but their cluster structure has some intriguing dissimilarities.

algebraic geometry

Audience: researchers in the topic

UC Davis algebraic geometry seminar