Twists of Gr(3,n) Cluster Variables as Double and Triple Dimer Partition Functions
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\"...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
24.05.2023
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Subjects | |
Online Access | Get full text |
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Summary: | 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). |
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DOI: | 10.48550/arxiv.2305.15531 |