UC Berkeley

Cluster algebras are a class of commutative rings with a remarkable combinatorial structure, introduced by Fomin and Zelevinsky. A cluster algebra has a distinguished set of generators, called cluster variables, which are grouped together into overlapping subsets called seeds. This dissertation is concerned with the cluster algebra structure of coordinate rings of open positroid varieties in the Grassmannian. Open positroid varieties are projections of open Richardson varieties from the full flag variety to the Grassmannian. They were studied first by Lusztig and Rietsch in the context of total positivity, and then by Knutson--Lam--Speyer, who connected them to the combinatorics of the totally nonnegative Grassmannian developed by Postnikov. Open positroid varieties are smooth, irreducible, and stratify the Grassmannian; open Schubert varieties are a special case.

Seminal work of Scott established that the homogeneous coordinate ring of the Grassmannian is a cluster algebra, and moreover that Postnikov's plabic graphs for the Grassmannian give seeds for this cluster algebra. Postnikov defined plabic graphs not just for the Grassmannian but for all positroid varieties. Accordingly, experts long believed that the coordinate ring of any open positroid variety is also a cluster algebra, with seeds given by plabic graphs. In Chapter 3, which is joint work with Khrystyna Serhiyenko and Lauren Williams, we prove this in the case of open Schubert varieties in the Grassmannian. Work of Leclerc on Richardson varieties in the full flag variety implies that the coordinate rings of these varieties are cluster algebras, but does not give any explicit descriptions of seeds. We show that Postnikov's plabic graphs give seeds in this cluster algebra. For skew Schubert varieties, we show that Leclerc's cluster algebra is given by relabeled plabic graphs, whose boundary vertices are permuted.

Shortly following my work with Serhiyenko and Williams, Galashin--Lam showed that Postnikov's graphs give a cluster algebra structure on coordinate rings of arbitrary positroid varieties using similar methods. In Chapter 4, which is joint work with Chris Fraser, we expand on this result to show that positroid varieties admit a number of different cluster structures, with seeds given by relabeled plabic graphs. Along the way, we show that many positroid varieties are isomorphic, using a permuted version of the Muller--Speyer twist map. We conjecture that all of these distinct cluster structures differ only by rescaling, and prove this conjecture for open Schubert varieties. This enlarges the class of combinatorially well-understood seeds for positroid varieties, which provides additional tools to further study the cluster structure on positroid varieties.