## 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.

Original language | English |
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Pages (from-to) | 3437-3527 |

Number of pages | 91 |

Journal | Duke mathematical journal |

Volume | 168 |

Issue number | 18 |

Early online date | 11 Nov 2019 |

DOIs | |

Publication status | Published - Nov 2019 |