A topology on the poset of quiver mutation classes

Abstract

To better understand mutation-invariant and hereditary properties of quivers (and more generally skew-symmetrizable matrices), we have constructed a topology on the set of all mutation classes of quivers which we call the mutation class topology. This topology is the Alexandrov topology induced by the poset structure on the set of mutation classes of quivers from the partial order of quiver embedding. The closed sets of our topology -- equivalently, the lower sets of the poset -- are in bijective correspondence with mutation-invariant and hereditary properties of quivers. We show that this space is strictly $T_0$, connected, non-Noetherian, and that every open set is dense. We close by providing open questions from cluster algebra theory in the setting of the mutation class topology and some directions for future research.