Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster
algebras. They are useful for computing refined Donaldson-Thomas invariants,
constructing twist automorphisms and proving the existence of theta bases and
generic bases. We survey recent progress on their existence and properties and
give a representation-theoretic proof of Greg Muller's theorem stating that
full subquivers inherit maximal green sequences. In the appendix, Laurent
Demonet describes maximal chains of torsion classes in terms of bricks
generalizing a theorem by Igusa.Comment: 15 pages, submitted to the proceedings of the ICRA 18, Prague,
comments welcome; v2: misquotation in section 6 corrected; v3: minor changes,
final version; v4: reference to Jiarui Fei's work added, post-final version;
v4: formulation of Remark 4.3 corrected; v5: misquotation of Hermes-Igusa's
2019 paper corrected; v5: reference to Kim-Yamazaki's paper adde