Twists of Gr ( 3 , n ) Cluster Variables as Double and Triple Dimer Partition Functions
Year: 2024
Type: article
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 Plücker coordinates, and give generating functions for their images under the twist map - a cluster algebra automorphism introduced in [1]. 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 in [13]), respectively. These connections shed light on a conjecture appearing in [3], extend the concept of web duality introduced in [9], and more broadly make headway on understanding Grassmannian cluster algebras for Gr ( 3 , n ) .
Source: Algebraic Combinatorics
DOI: 10.5802/alco.376
Authors: Moriah Elkin, Gregg Musiker, Kayla Wright
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4403939426