Graph Theoretical Cluster Expansion Formulas

Gregg Musiker

In this talk, I will discuss several examples of cluster algebras which have graph theoretic formulas for cluster expansions. These include cluster algebras of classical type with bipartite seeds, joint work with Jim Propp on rank 2 cluster algebras of affine type, and joint work with Ralf Schiffler for cluster algebras arising from triangulated surfaces.

In each of these cases, we explicitly construct a family of graphs such that cluster variables can be written as positive Laurent polynomials where a weighted enumeration of their perfect matchings encodes the numerator, while decompositions of the graphs correspond to the denominator.