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arXiv · 1503.07945

Semi-invariant pictures and two conjectures on maximal green sequences

Thomas Brüstle·Stephen Hermes·Kiyoshi Igusa·Gordana Todorov
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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 $Q$.

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Thomas Brüstle, Stephen Hermes, Kiyoshi Igusa, Gordana Todorov. 2015-03-27. Semi-invariant pictures and two conjectures on maximal green sequences. https://arxiv.org/abs/1503.07945

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