Manchester Geometry, Topology and Mathematical Physics Seminar 2021/2022

Cluster algebras, introduced by Fomin and Zelevinsky in 2000, are commutative rings with distinguished generators called cluster variables which are grouped into overlapping sets called clusters. Each pair of clusters is connected by a sequence of combinatorial procedures called mutations. These relations can be visualized with the exchange graph, which has a vertex for each cluster and an edge for each mutation. When A is a cluster algebra with finitely many clusters, the exchange graph of A is the 1-skeleton of a polytope called the generalized associahedron. The faces of the generalized associahedron are indexed by all subclusters. We define the superunital region S(A) to be the space of ring homomorphisms from A to the real numbers which send each cluster variable to a number equal to or greater than 1. We show that there is a homemorphism from S(A) to the generalized associahedron of A which sends each subcluster face to the corresponding polyhedral face. In particular, like the generalized associahedron, the region S(A) decomposes into faces indexed by subclusters. As an application, we prove that there are finitely many positive integral Dynkin-type friezes. This talk is based on work in progress with Greg Muller.