[Paper Review] The existence of a maximal green sequence is not invariant under quiver mutation

Greg Muller|arXiv (Cornell University)|Mar 16, 2015

Algebraic structures and combinatorial models4 citations

TL;DR

This paper demonstrates that the existence of a maximal green sequence is not invariant under quiver mutation by constructing a counterexample: a quiver (Q2,2,3) that admits no maximal green sequence, despite being mutation-equivalent to a quiver that does. Using scattering diagrams from the Gross-Hacking-Keel-Kontsevich theory, the authors prove that if a quiver admits a maximal green sequence, so must every induced subquiver—contradicted by Q2,2,3, which has an induced subquiver admitting such a sequence. The key result refutes a long-standing conjecture in cluster algebra theory.

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.

Motivation & Objective

• To disprove the conjecture that the existence of a maximal green sequence is invariant under quiver mutation.
• To establish that maximal green sequences are not preserved under mutation equivalence by constructing a counterexample.
• To demonstrate that the property of admitting a maximal green sequence is not inherited by all mutation-equivalent quivers.
• To use scattering diagrams and the sign coherence theorem to analyze the behavior of g-vectors and mutation sequences in cluster algebras.
• To clarify the limitations of maximal green sequences in relation to cluster algebra structure and upper cluster algebras.

Proposed method

• The authors use the scattering diagram framework developed by Gross-Hacking-Keel-Kontsevich to analyze wall structures in the space of g-vectors.
• They apply the mutation rule for scattering diagrams, showing how walls transform under quiver mutation via piecewise linear maps Gk and Rk.
• They prove that a finite transverse path in a scattering diagram corresponds to a maximal green sequence, with path direction indicating mutation order.
• They analyze the stereographic projection of the scattering diagram D(Q2,2,3) to visualize wall configurations and chamber connectivity.
• They show that in Q2,2,3, any path from the innermost chamber to the exterior is trapped by a purple half-space after crossing a single wall, preventing finite transverse paths.
• They use the fact that induced subquivers inherit the property of admitting a maximal green sequence if the full quiver does, to derive a contradiction when applied to Q2,2,3.

Experimental results

Research questions

• RQ1 Does the existence of a maximal green sequence remain invariant under quiver mutation?
• RQ2 Can a quiver that is mutation-equivalent to one with a maximal green sequence fail to admit such a sequence itself?
• RQ3 What role do scattering diagrams play in determining the existence of maximal green sequences?
• RQ4 How do wall structures in scattering diagrams obstruct or allow finite transverse paths corresponding to maximal green sequences?
• RQ5 Is the property of admitting a maximal green sequence preserved under taking induced subquivers?

Key findings

• The quiver Q2,2,3, depicted in Figure 1, does not admit a maximal green sequence.
• Despite being mutation-equivalent to a quiver that does admit a maximal green sequence, Q2,2,3 fails to support such a sequence.
• The scattering diagram D(Q2,2,3) contains an infinite number of walls in a purple half-space, which trap any finite transverse path after the first crossing.
• The path from the innermost chamber to the exterior chamber cannot be completed without crossing walls from the convex side, violating the requirement for a maximal green sequence.
• The induced subquiver on vertices 1,2,3 of Q2,2,3 does admit a maximal green sequence, contradicting the invariance under mutation.
• The result refutes Conjecture 1.3.3, which posited that maximal green sequence existence is preserved under mutation equivalence.