Lengths of maximal green sequences for tame path algebras

TL;DR

This paper investigates the maximum lengths of maximal green sequences for certain tame quivers, showing independence from orientation and providing explicit formulas and a counting program.

Contribution

It explicitly determines the maximal length of green sequences for tame path algebras of types D and E, and introduces a program to count sequences by length.

Findings

Maximal length is independent of quiver orientation.

Explicit formulas for maximal lengths are provided.

A program to count all sequences by length is developed.

Abstract

In this paper, we study the maximal length of maximal green sequences for quivers of type $\widetilde{\mathbf{D}}$ and $\widetilde{\mathbf{E}}$ by using the theory of tilting mutation. We show that the maximal length does not depend on the choice of the orientation, and determine it explicitly. Moreover, we give a program which counts all maximal green sequences by length for a given Dynkin/extended Dynkin quiver.