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

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 = Gr n−k (C n ), as well as the mirror dual Landau-Ginzburg model (X ○ , W ∶X ○ → C), whereX ○ is the complement of a particular anti-canonical divisor in a Langlands dual GrassmannianX = Gr k ((C n ) * ), and the superpotential W has a simple expression in terms of Plücker coordinates [MR13]. Grassmannians simultaneously have the structure of an A-cluster variety and an X -cluster variety [Sco06, Pos]; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps [FZ02, FG06]. Given a plabic graph or, more generally, a cluster seed G, we consider two associated coordinate systems: a network orare the open positroid varieties in X andX, respectively. To each X -cluster chart Φ G and ample 'boundary divisor' D in X ∖ X ○ , we associate a Newton-Okounkov body ∆ G (D) in R k(n−k) , which is defined as the convex hull of rational points; these points are obtained from the multi-degrees 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 A-cluster 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 "tropicalising". Our first main result is that the Newton-Okounkov bodies ∆ G (D) and the polytopes obtained by tropicalisation 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 that the cluster seed G corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians [FW04].

Full-Rank Valuations and Toric Initial Ideals

“…In this section we apply Theorem 1 from §3 to the valuation v G defined by Rietsch-Williams for Grassmannians using the cluster structure and Postnikov's plabic graphs in [RW17]. The same combinatorial objects are used in [BFF + 18] to define weight vectors.…”

“…We recall plabic graphs in the Appendix A below. For every plabic graph G (or more generally every seed) for Gr k (C n ) in [RW17] they define a valuation v G : A k,n \ {0} → Z d where d := k(n − k) = dim Gr k (C n ). The images of Plücker coordinates v G (p J ) can be computed using the combinatorics of the plabic graph G. Please consider Appendix B for the precise definition of the valuation and how to compute it.…”

“…Example 6. Let G be one of the plabic graphs G 1 , G 2 for Gr(3, C 6 ) in [RW17,§8]. The associated Newton-Okounkov polytope is not integral and the value semi-group S(A 3,6 , v G ) is generated by {v G (p ijk )|1 ≤ i < j < k ≤ n} and one more element v G (f ), where f ∈ A 3,6 and deg(f ) = 2.…”

“…Grassmannians and valuations from plabic graphs. We consider a class of valuations on the homogeneous coordinate rings of Grassmannians defined in [RW17]. In the context of cluster algebras and cluster duality for Grassmannians, they associate a full-rank valuation v G to every plabic graph G with certain properties [Pos06].…”

“…Related to [RW17], in [BFF + 18] the authors associate plabic weight vectors w G for C[p J ] J to the same plabic graphs G (see Definition 14). They study these weight vector for Gr 2 (C n ) and Gr 3 (C 6 ) and show that, in these cases, they lie in the tropical Grassmannian [SS04], i.e.…”