Melissa Sherman-Bennett: Plabic graphs and cluster structures on positroid varieties

Abstract

Open positroid varieties are smooth irreducible subvarieties of the Grassmannian, which can be naturally defined using "positively realizable" matroids (positroids, for short). They were first introduced by Knutson, Lam, and Speyer, motivated by work of Postnikov on the totally nonnegative (real) Grassmannian and positroid cells. Open positroid varieties are indexed by a number of combinatorial objects, including families of plabic (i.e. planar bicolored) graphs. I'll discuss some algebraic information plabic graphs give us about open positroid varieties. Together with K. Serhiyenko and L. Williams, we showed that plabic graphs for an open Schubert variety V (a special case of open positroid varieties) give seeds for a cluster algebra structure on the homogeneous coordinate ring of V. Among other things, this implies that plabic graphs give positivity tests for elements of V. Our work generalizes a result of Scott on the Grassmannian, and confirms a longstanding folklore conjecture on Schubert varieties; it was later generalized to arbitrary positroid varieties by Galashin and Lam. I'll also discuss recent work with C. Fraser, in which we show that relabeled plabic graphs also give seeds for a cluster algebra structure on coordinate rings of open positroid varieties, uncovering another source for positivity tests. No knowledge of cluster algebras will be assumed in the talk.