Mikhail Gorsky (University of Vienna), A structural view of maximal green sequences, I and II.

Abstract:

Maximal green sequences were first introduced by Keller in the studies of connections between cluster algebras and quantum dilogarithm identities. The talk concerns the structure of the set of all maximal green sequences of a finite-dimensional algebra. There is a natural equivalence relation on this set, which, as I will explain, can be interpreted in several different ways, underscoring its significance. I will define several partial orders on the equivalence classes of maximal green sequences, analogous to the partial orders on silting complexes and generalizing higher Stasheff-Tamari orders on triangulations of three-dimensional cyclic polytopes. We conjecture that these partial orders are in fact equal, just as the orders in the silting case have the same Hasse diagram. This can be seen as a refined and more widely applicable version of the no-gap conjecture of Brüstle, Dupont, and Perotin. I will sketch the proof of the conjecture for Nakayama algebras. If time permits, I will also discuss two-dimensional higher Bruhat orders and Cambrian maps in the context of posets of maximal green sequences.

The talk is based on joint work with Nicholas Williams (arXiv:2301.08681).