Mixed Dimer Configuration Model in Type D Cluster Algebras

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.