Twists of Plücker Coordinates as Dimer Partition Functions

Abstract

The homogeneous coordinate ring of the Grassmannian Gr k,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plucker coordinates. We introduce a twist map on Gr k,n , related to the Berenstein–Fomin–Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plucker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plucker coordinate. We also relate the twist map to a maximal green sequence.