Cluster duality and mirror symmetry for the Grassmannian

In joint work with Konstanze Rietsch, we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. From a given plabic graph G we have two coordinate systems: we have a positive chart for our A-model Grassmannian, and we have a cluster chart for our B-model (Landau-Ginzburg model) Grassmannian. On the A-model side, we use the positive chart to associate a corresponding Newton-Okounkov (A-model) polytope. On the B-model side, we use the cluster chart to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a B-model polytope. Our main result is that these two polytopes coincide

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