Mixed Dimer Configuration Model in Type D Cluster Algebras

TL;DR

This paper introduces a combinatorial mixed dimer configuration model for type D cluster algebras, providing a graph-theoretic method to compute F-polynomials and g-vectors for acyclic quivers.

Contribution

It develops a novel mixed dimer configuration model that unifies dimer and double dimer configurations to compute key algebraic invariants in type D cluster algebras.

Findings

Graph-theoretic recipe for F-polynomials

Method to determine coefficients of monomials

Weighting scheme for g-vectors

Abstract

We give a combinatorial model for F-polynomials and g-vectors for type D cluster algebras where the associated quiver is acyclic. Our model utilizes a combination of dimer configurations and double dimer configurations which we refer to as mixed dimer configurations. In particular, we give a graph theoretic recipe that describes which monomials appear in such F-polynomials, as well as a graph theoretic way to determine the coefficients of each of these monomials. In addition, we give a weighting on our mixed dimer configuration model that gives the associated g-vector. To prove this formula, we use a combinatorial formula due to Thao Tran and provide explicit bijections between her combinatorial model and our own.