The Existence of a Maximal Green Sequence is not Invariant under Quiver Mutation

This note provides a specific quiver which does not admit a maximal green sequence, and which is mutation-equivalent to a quiver which does admit a maximal green sequence. 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. 1 Quivers and maximal green sequences 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. Figure 1: The counterexample quiver 1.1 Quiver mutation 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. the electronic journal of combinatorics 23(2) (2016), #P2.47 1 1. For each pair of arrows i → k → j through the vertex k, add an arrow i→ j. 2. Reverse the orientation of every arrow incident to k. 3. Cancel any directed 2-cycles in pairs. Q k (1) k