Minimal length maximal green sequences

Abstract

Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine what lengths are achieved by the maximal green sequences of a quiver. We combine the combinatorics of surface triangulations and the basics of scattering diagrams to address this problem. Our main result is a formula for the length of minimal length maximal green sequences of quivers defined by triangulations of an annulus or a punctured disk.

Building maximal green sequences via component preserving mutations

“…There has been recent interest in finding maximal green sequences of minimal possible length for a given quiver [11,21]. We will now show how minimal length maximal green sequences can be constructed with component preserving mutations.…”

“…We will now show how minimal length maximal green sequences can be constructed with component preserving mutations. In additional to being a natural question to ask about maximal green sequences, it has been observed by Garver, McConville, and Serhiyenko that the minimal possible length of a maximal green sequence may be related to derived equivalence of cluster tilted algebras (see [21,Question 10.1]).…”

“…The following result is a component preserving generalization of [21,Proposition 4.4] which considers the direct sum case.…”

“…Consider any maximal green sequence τ for Q. By [21,Theorem 3.3] it follows that for each 1 ≤ i ≤ there is a subsequence of mutations in τ at vertices in Q i which is a maximal green sequence of Q i . This means τ must mutate at vertices of Q i at least L i times for each 1 ≤ i ≤ .…”

Maximal green sequences for cluster-tilted algebras of finite representation type

“…Upper bound for lengths of MGSs. For many cluster-tilted algebras of finite representation type the minimum length of a maximal green sequence has been computed [18], [23]. In particular we have the following.…”