Twists of Gr(3,n) Cluster Variables as Double and Triple Dimer Partition Functions

Abstract

We give a combinatorial interpretation for certain cluster variables in Grassmannian cluster algebras in terms of double and triple dimer configurations. More specifically, we examine several Gr(3,n) cluster variables that may be written as degree two or degree three polynomials in terms of Plucker coordinates, and give generating functions for their images under the twist map - a cluster algebra automorphism introduced in [BFZ96]. The generating functions range over certain double or triple dimer configurations on an associated plabic graph, which we describe using particular noncrossing matchings or webs (as in [Kup96]), respectively. These connections shed light on a conjecture appearing in [CDH+22], extend the concept of web duality introduced in [FLL19], and more broadly make headway on understanding Grassmannian cluster algebras for Gr(3,n).