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.

arXiv comment49 pages, 46 figures, updated to address properties of the poset as well as draw a direct connection to representation theory