Newton-Okounkov bodies and cluster duality for Grassmannians

We use the A and X-cluster structures on the Grassmannian to exhibit a polytopal manifestation of mirror symmetry for Grassmannians. From a given cluster seed we obtain both an X-cluster chart and an A-cluster chart for the Grassmannian. We use the X-cluster chart and a naturally defined valuation to construct a Newton-Okounkov body, defined as the convex hull of points. Meanwhile we use the A-cluster to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a polytope, defined by inequalities. We prove that the Newton-Okounkov body and the superpotential polytope coincide. In the case that our A-cluster 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 Fulton-Woodward, this formula has an interpretation in terms of quantum cohomology. This is joint work with Konstanze Rietsch.

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