Semi-invariant pictures and two conjectures on maximal green sequences

Thomas Brüstle; Stephen Hermes; Kiyoshi Igusa; Gordana Todorov

Semi-invariant pictures and two conjectures on maximal green sequences

Representation Theory 2015-10-12 v2

Authors: Thomas Brüstle , Stephen Hermes , Kiyoshi Igusa , Gordana Todorov

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 $Q$ .

Keywords

maximal green sequences semi-invariant pictures quiver mutation acyclic valued quiver rotation lemma tame type reddening sequences

Cite

@article{arxiv.1503.07945,
  title  = {Semi-invariant pictures and two conjectures on maximal green sequences},
  author = {Thomas Brüstle and Stephen Hermes and Kiyoshi Igusa and Gordana Todorov},
  journal= {arXiv preprint arXiv:1503.07945},
  year   = {2015}
}

Comments

24 pages, 7 figures

Semi-invariant pictures and two conjectures on maximal green sequences

Abstract

Keywords

Cite

Comments

Related papers