Mixed Dimer Configuration Model in Type $D$ Cluster Algebras II: Beyond the Acyclic Case

TL;DR

This paper extends a combinatorial dimer configuration model for type D cluster algebras to include quivers with oriented cycles, providing a graph-theoretic method to compute $F$-polynomials and their coefficients.

Contribution

It introduces a new graph-theoretic recipe for $F$-polynomials in type D cluster algebras with cycles, and establishes a bijection with module dimension vectors.

Findings

Graph-theoretic description of monomials in $F$-polynomials.

Method to determine coefficients of monomials.

Bijection between dimer configurations and module dimension vectors.

Abstract

This is a sequel to the second and third author's Mixed Dimer Configuration Model in Type $D$ Cluster Algebras where we extend our model to work for quivers that contain oriented cycles. Namely, we extend a combinatorial model for $F$-polynomials for type $D_n$ using dimer and double 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. To prove this formula, we provide an explicit bijection between mixed dimer configurations and dimension vectors of submodules of an indecomposable Jacobian algebra module.