The existence of a maximal green sequence is not invariant under quiver mutation

TL;DR

This paper demonstrates that the property of having a maximal green sequence is not preserved under quiver mutation by providing a counterexample and using scattering diagrams to analyze the property.

Contribution

It introduces a counterexample quiver that challenges the invariance of maximal green sequences under mutation, utilizing scattering diagrams for the proof.

Findings

A quiver without a maximal green sequence can be mutation-equivalent to one with such a sequence.

Maximal green sequences are not invariant under quiver mutation.

Scattering diagrams are effective in analyzing properties related to quiver mutations.

Abstract

This note provides a quiver which does not admit a maximal green sequence, but which is mutation-equivalent to a quiver which does admit a maximal green sequence. The proof uses the `scattering diagrams' of Gross-Hacking-Keel-Kontsevich to show that a maximal green sequence for a quiver determines a maximal green sequence for any induced subquiver.