Maximal Green Sequences for Cluster Algebras Associated to Closed Orbifolds

TL;DR

This paper investigates the existence of maximal green sequences in cluster algebras derived from orbifolds, providing triangulations for certain surfaces and conditions for such sequences to exist.

Contribution

It extends the study of maximal green sequences from surfaces to orbifolds, offering new triangulations and criteria for their existence.

Findings

Provided triangulations for orientable surfaces of genus n with orbifold points and punctures.

Determined conditions under which these surfaces admit maximal green sequences.

Constructed explicit maximal green sequences when they exist.

Abstract

It is known that the existence of a maximal green sequence for a quiver associated to surfaces is equivalent to the equality of the cluster algebra and upper cluster algebra generated by the quiver. This paper makes the first steps in investigating this behavior in the generalised case of cluster algebras from orbifolds; determining when such surfaces admit a diagram with a maximal green sequence. Specifically, we will provide a triangulation for the orientable surfaces of genus $n$ with an arbitrary number of orbifold points and arbitrary number of punctures, determine when it has a maximal green sequence, and construct one if it exists.