On Maximal Green Sequences

TL;DR

This paper investigates the properties and finiteness of maximal green sequences in quiver mutation theory, providing new results on their enumeration and characteristics across different classes of quivers.

Contribution

It proves the finiteness of maximal green sequences for specific quiver classes and explores their possible counts and lengths, advancing understanding in representation theory.

Findings

Finiteness of maximal green sequences for cluster finite, affine, and small acyclic quivers.

Results on the possible numbers of maximal green sequences.

Results on the possible lengths of maximal green sequences.

Abstract

Maximal green sequences are particular sequences of quiver mutations appearing in the context of quantum dilogarithm identities and supersymmetric gauge theory. Interpreting maximal green sequences as paths in various natural posets arising in representation theory, we prove the finiteness of the number of maximal green sequences for cluster finite quivers, affine quivers and acyclic quivers with at most three vertices. We also give results concerning the possible numbers and lengths of these maximal green sequences.