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Newton-Okounkov bodies, cluster duality and mirror symmetry for Grassmannians

Research output: Contribution to journalArticle

Konstanze Rietsch, Lauren Williams

Original languageEnglish
Pages (from-to)3437-3527
Number of pages91
JournalDuke mathematical journal
Volume168
Issue number18
Early online date11 Nov 2019
DOIs
Accepted/In press9 Apr 2019
E-pub ahead of print11 Nov 2019
Published2019

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Abstract

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.

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