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 associated to most marked surfaces. We develop a procedure to find maximal green sequences for cluster quivers associated to an arbitrary triangulation of closed higher genus marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with at least one boundary component has a maximal green sequence.
Highlights
Cluster algebras were introduced by Fomin and Zelevinsky in [14]
One very important property of a quiver associated to a cluster algebra is whether or not it has a maximal green sequence
For all other quivers from marked surfaces it is known that A = U if and only if there exists a quiver with a maximal green sequence
Summary
Cluster algebras were introduced by Fomin and Zelevinsky in [14]. Cluster algebras have become an important tool in the study of many areas of mathematics and mathematical physics. For all other quivers from marked surfaces it is known that A = U if and only if there exists a quiver with a maximal green sequence. For any triangulation of Σ there exists a maximal green sequence for the associated quiver. The existence of maximal green sequences for special triangulation of various marked surfaces has been shown in many papers [1, 5, 6]. It is known that all finite mutation type quivers arise from triangulations of surfaces except for the rank 2 case and 11 exceptional cases Among these exceptional cases it has been shown that there exists a quiver with a maximal green sequence for all but X7 [1].
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