A survey on maximal green sequences

Abstract

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.

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 added