King's College London

In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian X D Gr _n-k.C ⁿ/, as well as the mirror dual Landau-Ginzburg model .X ^{L ı}; W W X ^{L ı} ! C/, where X ^{L ı} is the complement of a particular anticanonical divisor in a Langlands dual Grassmannian X ^L D Gr _k..C ⁿ/*/ and the superpotential W has a simple expression in terms of Plücker coordinates. Grassmannians simultaneously have the structure of an A-cluster variety and an X-cluster variety; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps. Given a plabic graph or, more generally, a cluster seed G, we consider two associated coordinate systems: a network or X-cluster chart Φ _G W .C*/ ^k.n-k/ ! X ^ı and a Plücker cluster or A-cluster chart Φ ^_ _G W .C*/ ^k.n-k/ ! X ^{L ı}. Here X ^ı and X ^{L ı} are the open positroid varieties in X and X ^L , respectively. To each X-cluster chart Φ _G and ample boundary divisor D in XnX ^ı, we associate a Newton-Okounkov body Δ _G.D/ in R ^k.n-k/, which is defined as the convex hull of rational points; these points are obtained from the multidegrees of leading terms of the Laurent polynomials Φ* _G.f / for f on X with poles bounded by some multiple of D. On the other hand, using the Acluster chart Φ ^_ _G on the mirror side, we obtain a set of rational polytopes-described in terms of inequalities-by writing the superpotential W as a Laurent polynomial in the A-cluster coordinates and then tropicalizing. Our first main result is that the Newton-Okounkov bodies Δ _G.D/ and the polytopes obtained by tropicalization on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of) these Newton-Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton-Okounkov bodies, in the case in which the cluster seed G corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians.