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.
Highlights
This provides a counterexample to the conjecture that the existence of maximal green sequences is invariant under quiver mutation
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
The purpose of this paper is to prove that the quiver shown in Figure 1 is a counterexample to the conjecture that the existence of maximal green sequences is invariant under quiver mutation
Summary
A quiver is a finite directed graph without loops or 2-cycles. A quiver Q may be mutated at a vertex k to produce a new quiver μk(Q) in three steps. Mutation equivalence is an equivalence relation, because mutating at the same vertex twice in a row returns to the original quiver. We use the Sign Coherence Theorem for quivers to give an elementary definition of g-vectors, via g-seeds. Every vertex in an initial g-seed is green. A g-seed may be mutated at a vertex k which is green or red. After any sequence of mutations of an initial g-seed, every vertex in the resulting g-seed is either green or red. A g-seed which is mutation equivalent to an initial g-seed may be mutated at an arbitrary sequence of vertices. We have used the Sign Coherence Theorem to simplify the recursive identities (6.12) and (6.13) in [FZ07]; we have removed the explicit dependence on a choice of initial seed
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