Maximal green sequences for quivers of finite mutation type

TL;DR

This paper proves mutation invariance of maximal green sequences for finite mutation type quivers, providing explicit methods to find them for various classes of cluster quivers, including those from marked surfaces.

Contribution

It establishes mutation invariance of maximal green sequences for finite mutation type quivers and develops explicit procedures for their construction.

Findings

Maximal green sequences exist for all finite mutation type quivers except specific exceptions.

Explicit procedures are provided for triangulations of closed marked surfaces with at least two punctures.

Maximal green sequences are computed for exceptional finite mutation type quivers.

Abstract

In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers of finite mutation type. In particular, we show that a mutation finite cluster quiver has a maximal green sequence unless it arises from a once-punctured closed marked surface, or one of the two quivers in the mutation class of X7. We develop a procedure to explicitly find maximal green sequences for cluster quivers associated to arbitrary triangulations of closed marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with boundary has a maximal green sequence. We also compute explicit maximal green sequences for exceptional quivers of finite mutation type.