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

TL;DR

This paper provides a combinatorial interpretation of certain Grassmannian cluster variables using double and triple dimer configurations, linking cluster algebra automorphisms to web and matching structures to deepen understanding of Gr(3,n).

Contribution

It introduces a novel combinatorial framework connecting cluster variables to dimer configurations and webs, advancing the understanding of Grassmannian cluster algebras for Gr(3,n).

Findings

Generated functions correspond to specific dimer configurations.

Extended web duality concepts in the context of Grassmannian cluster algebras.

Provided insights into a conjecture by Cheung et al.

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 Pl\"ucker coordinates, and give generating functions for their images under the twist map - a cluster algebra automorphism introduced in work of Berenstein-Fomin-Zelevinsky. The generating functions range over certain double or triple dimer configurations on an associated plabic graph, which we describe using particular non-crossing matchings or webs (as defined by Kuperberg), respectively. These connections shed light on a recent conjecture of Cheung et al., extend the concept of web duality introduced in a paper of Fraser-Lam-Le, and more broadly make headway on understanding Grassmannian cluster algebras for Gr(3,n).