2015/07/28 by Rietsch, Konstanze, Williams, Lauren
#13F60 #14J33 #14M15 #52B20 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1507.07817
In this article 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. For our A-model, we consider the Grassmannian \mathbb X=Grn-k(ℂn). The B-model is a Landau-Ginzburg model (\check\mathbb X^∘, Wq:\check\mathbb X^∘ → ℂ), where \check\mathbb X^∘ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian \check\mathbb X = Grk((ℂn)^*), and the superpotential Wq has a simple expression in terms of Plücker coordinates, see [MarshRietsch]. From a given plabic graph G we obtain two coordinate systems: using work of Postnikov and Talaska we have a positive chart ΦG:(ℂ^*)k(n-k)→ \mathbb X in our A-model, and using work of Scott we have a cluster chart ΦG\vee:(ℂ^*)k(n-k)→ \check\mathbb X in our B-model. To each positive chart ΦG and choice of positive integer r, we associate a polytope NOGr, which we construct as the convex hull of a set of integer lattice points. This polytope is an example of a Newton-Okounkov polytope associated to the line bundle \mathcal O(r) on \mathbb X. On the other hand, using the cluster chart ΦG\vee and the same positive integer r, we obtain a polytope QGr -- described in terms of inequalities -- by "tropicalizing" the composition Wtr∘ ΦG\vee. Our main result is that the polytopes NOGr and QGr coincide.