Combinatorics of Quiver Mutations

Abstract:

A cluster algebra is a ring defined by combinatorial data: a quiver (directed graph) viewed up to mutation equivalence. Mutations are a particular kind of quiver transformation. A central open question in cluster combinatorics is that of detecting whether two given quivers are mutation equivalent. We approach this problem from two directions. First, we construct new arbitrarily long mutation cycles, i.e., sequences of mutations transforming a quiver into itself. These cycles give a barrier to greedy algorithms for deciding mutation equivalence. We show that a plethora of such cycles can be constructed using quivers that possess reddening sequences. Second, we introduce new mutation invariants that yield necessary conditions for mutation equivalence. In many cases, these invariants provide a quick way to establish that two quivers are mutation inequivalent, or that a given quiver cannot be mutated into an acylic one.