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

TL;DR

We address the problem of constructing explicit maximal green sequences for a broad class of quivers by introducing the $\mathcal{Q}^N$ quivers, comprising $N$ vertical chains with inter-chain arrows. A partial order on vertices and an associated mutation sequence yield a main theorem: there exists an explicit maximal green sequence for any $\mathcal{Q}^N$, with the proof built by induction on the number of vertical chains and reduction to the $N=1,2$ cases. The authors then show that several important families — finite connected full subquivers from Hernandez–Leclerc quivers, trees of oriented cycles, and quivers mutation-equivalent to type $A$ or $D$ orientations — are all instances of $\mathcal{Q}^N$, enabling explicit MG sequences for these cases and unifying prior results. This framework resolves open problems (notably for trees of oriented cycles) and provides a versatile method to obtain MG sequences across diverse quiver families with potential applications to cluster algebras and monoidal categorifications.

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.

Figures (25)

• Figure 3: (a) is the mutation at $v^{i}_{a_1}$ in $\widehat{Q^2}$, where $a_{1}=1$, $a_1<a_2-1$; (b) is the mutation at $v^{i}_{a_1}$ in $\widehat{Q^2}$, where $a_{1}=1$, $a_1=a_2-1$.
• Figure 5: In type $B_2$, quiver $G$ (left) and a finite full subquiver $\bar{G}$ of $G$ (right).
• Figure 6: A tree of oriented cycles.
• Figure 9: A quiver in $\mu^A$.
• Figure : (a)

Theorems & Definitions (34)

• Definition 3.1
• Remark 3.1
• Definition 3.2
• Example 3.1
• Definition 3.3
• Remark 3.2
• Lemma 3.2
• proof
• Remark 3.3
• Theorem 3.3