Properties of minimal mutation-infinite quivers

TL;DR

This paper investigates the properties of minimal mutation-infinite quivers, demonstrating their structural features, existence of maximal green sequences, and implications for associated cluster algebras.

Contribution

It establishes that all minimal mutation-infinite quivers of rank 4 or higher are Louise and possess maximal green sequences, linking these properties to cluster algebra structures.

Findings

Rank 4 or higher minimal mutation-infinite quivers are Louise and have maximal green sequences.

In rank 3, at most 6 quivers in a mutation class admit a maximal green sequence.

For each rank 4 minimal mutation-infinite quiver, a finite subgraph with quivers having maximal green sequences exists.

Abstract

We study properties of minimal mutation-infinite quivers. In particular we show that every minimal-mutation infinite quiver of at least rank 4 is Louise and has a maximal green sequence. It then follows that the cluster algebras generated by these quivers are locally acyclic and hence equal to their upper cluster algebra. We also study which quivers in a mutation-class have a maximal green sequence. For any rank 3 quiver there are at most 6 quivers in its mutation class that admit a maximal green sequence. We also show that for every rank 4 minimal mutation-infinite quiver there is a finite connected subgraph of the unlabelled exchange graph consisting of quivers that admit a maximal green sequence.