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GASC Seminar "Combinatorics of Cluster Structures in Schubert Varieties" by Melissa Sherman-Bennett (Harvard)

Abstract:  The (affine cone over the) Grassmannian is a prototypical example of a variety with "cluster structure"; that is, its coordinate ring is a cluster algebra. Scott (2006) gave a combinatorial description of this cluster algebra in terms of Postnikov's plabic graphs. It has been conjectured essentially since Scott's result that Schubert varieties also have a cluster structure with a description in terms of plabic graphs.

I will discuss recent work with K. Serhiyenko and L. Williams proving this conjecture. The proof uses a result of Leclerc, who shows that many Richardson varieties in the full flag variety have cluster structure using cluster-category methods, and a construction of Karpman to build plabic graphs for each Schubert variety. Time permitting, I will also discuss our results on cluster structures on a larger class of positroid varieties, which involve the combinatorics of "generalized" plabic graphs.

Monday, March 11, 2019 at 12:15pm to 1:15pm

511 Lake Hall

Event Type

Seminar

Departments

COS, Mathematics, Geometry, Algebra, Singularities, Combinatorics (GASC)

Sponsoring Organization

Mathematics Department

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