Maximal green sequences for $\mathcal{Q}^N$ quivers

TL;DR

This paper introduces a new class of quivers called $Q^N$ quivers, constructs maximal green sequences for them, and shows they encompass several important classes of quivers, resolving an open problem in the field.

Contribution

It defines $Q^N$ quivers and proves they include various known quivers, providing a unified framework for constructing maximal green sequences.

Findings

Constructed maximal green sequences for $Q^N$ quivers.

Unified several classes of quivers under $Q^N$ framework.

Resolved an open problem by Garver and Musiker.

Abstract

We introduce $\mathcal{Q}^N$ quivers and construct maximal green sequences for these quivers. We prove that any finite connected full subquiver of the quivers defined by Hernandez and Leclerc, arising in monoidal categorifications of cluster algebras, is a special case of $\mathcal{Q}^N$ quivers. Moreover, we prove that the trees of oriented cycles introduced by Garver and Musiker are special cases of $\mathcal{Q}^N$ quivers. This result resolves an open problem proposed by Garver and Musiker, providing a construction of maximal green sequences for quivers that are trees of oriented cycles. Furthermore, we prove that quivers that are mutation equivalent to an orientation of a type AD Dynkin diagram can also be recognized as special cases of $\mathcal{Q}^N$ quivers.