A structural view of maximal green sequences

TL;DR

This paper investigates the structure of maximal green sequences in finite-dimensional algebras, introducing new partial orders and conjecturing their equivalence, with proof provided for Nakayama algebras.

Contribution

It introduces a novel framework for understanding maximal green sequences through partial orders and conjectures their equivalence, extending the No-Gap Conjecture.

Findings

Defined an equivalence relation on maximal green sequences

Established three partial orders on equivalence classes

Proved the conjecture for Nakayama algebras

Abstract

We study 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 we show can be interpreted in several different ways, underscoring its significance. There are three partial orders on the equivalence classes, analogous to the partial orders on silting complexes and generalising the 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 Br\"ustle, Dupont, and Perotin. We prove our conjecture in the case of Nakayama algebras.