NewtonOkounkov bodies and cluster duality for Grassmannians
Connections for Women: Enumerative Geometry Beyond Numbers January 18, 2018  January 19, 2018
Location: SLMath: Eisenbud Auditorium
mirror symmetry
Grassmannian
cluster algebra
cluster variety
NewtonOkounkov body
Schubert calculus
7Williams
We use the A and Xcluster structures on the Grassmannian to exhibit a polytopal manifestation of mirror symmetry for Grassmannians. From a given cluster seed we obtain both an Xcluster chart and an Acluster chart for the Grassmannian. We use the Xcluster chart and a naturally defined valuation to construct a NewtonOkounkov body, defined as the convex hull of points. Meanwhile we use the Acluster to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a polytope, defined by inequalities. We prove that the NewtonOkounkov body and the superpotential polytope coincide. In the case that our Acluster consists of Plucker coordinates, we also give a formula for each lattice point of these polytopes in terms of Young diagrams; using a result of FultonWoodward, this formula has an interpretation in terms of quantum cohomology. This is joint work with Konstanze Rietsch.
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