Semi-invariant pictures and two conjectures on maximal green sequences

TL;DR

This paper proves two conjectures about maximal green sequences in quivers using semi-invariant pictures, establishing mutation order constraints and finiteness results for certain classes of quivers.

Contribution

It introduces a novel application of semi-invariant pictures to prove conjectures on maximal green sequences in acyclic and tame quivers.

Findings

Maximal green sequences must mutate at specific vertices in acyclic valued quivers with infinite type arrows.

Finitely many maximal green sequences exist for quivers obtained by mutation from tame acyclic valued quivers.

The Rotation Lemma and Mutation Formula are key tools in these proofs.

Abstract

We use semi-invariant pictures to prove two conjectures about maximal green sequences. First: if $Q$ is any acyclic valued quiver with an arrow $j\to i$ of infinite type then any maximal green sequence for $Q$ must mutate at $i$ before mutating at $j$. Second: for any quiver $Q'$ obtained by mutating an acyclic valued quiver $Q$ of tame type, there are only finitely many maximal green sequences for $Q'$. Both statements follow from the Rotation Lemma for reddening sequences and this in turn follows from the Mutation Formula for the semi-invariant picture for $Q$.