On the combinatorics of quivers, mutations and cluster algebra exchange graphs

Abstract

Over the last 20 years, cluster algebras have been widely studied, with numerous links to different areas of mathematics and physics. These algebras have a cluster structure given by successively mutating seeds, which can be thought of as living on some graph or tree. In this way one can use various combinatorial tools to discover more about these cluster structures and the cluster algebras themselves.

This thesis considers some of the combinatorics at play here. Mutation-finite quivers have been classified, with links to triangulations of surfaces and semi-simple Lie algebras, while comparatively little is known about mutation-infinite quivers. We introduce a classification of the minimal types of these mutation-infinite quivers before studying their properties. We show that these minimal mutation-infinite quivers admit a maximal green sequence and that the cluster algebras which they generate are equal to their related upper cluster algebras.

Automorphisms of skew-symmetric cluster algebras are known to be linked to automorphisms of their exchange graphs. In the final chapter we discuss how this idea can be extended to skew-symmetrizable cluster algebras by using the symmetrizing weights to add markings to the exchange graphs. This opens possible opportunities to study orbifold mapping class groups using combinatoric graph theory.