Many cluster structures on positroid varieties

La exposición es parte del Online Cluster Algebra Seminar, en cuyo comité organizador está involucrado Daniel Labardini.

Many cluster structures on positroid varieties

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 give a family of cluster structures on open Schubert (and more generally, positroid) varieties. Each of the cluster structures in this family has seeds given by face labels of relabeled plabic graphs, which are plabic graphs whose boundary is labeled by a permutation of 1, ..., n. For Schubert varieties, all cluster structures in this family quasi-coincide, meaning they differ only by rescaling by frozen variables and their cluster monomials coincide. In particular, all relabeled plabic graphs for a Schubert variety give rise to seeds in the "usual" cluster algebra structure on the coordinate ring. As part of our results, we show the "target" and "source" cluster structures on Schubert varieties quasi-coincide, confirming a conjecture of Muller and Speyer. One proof tool of independent interest is a permuted version of the Muller-Speyer twist map, which we use to prove many (open) positroid varieties are isomorphic.