An explicit description of the colored mutation class of $\widetilde{\mathbb{A}}_n$-quivers

TL;DR

The paper provides a complete, purely combinatorial classification of the $m$-colored mutation class of quivers of extended Dynkin type $\widetilde{\mathbb{A}}_{p,q}$. It introduces central cycles and the class $\mathcal{Q}^m_{p,q}$, proving that mutation-equivalence to $\widetilde{\mathbb{A}}_{p,q}$ is equivalent to belonging to $\mathcal{Q}^m_{p,q}$. This yields a constructive, mutation-closed framework that generalizes the $m=1$ case studied by Bastian and describes all quivers underlying $m$-cluster tilted algebras of type $\tilde{\mathbb{A}}_{p,q}$. The results provide a purely combinatorial method to obtain and study these mutation classes and their associated algebras, with consequences aligning with and extending existing classifications in the literature.

Abstract

This paper addresses the combinatorial structure of $m$-colored mutation classes. We provide an explicit and purely combinatorial description of the $m$-colored quivers that arise within the $m$-colored mutation class of a quiver of type $\mathbb{\widetilde{A}}$. Our results generalize and extend existing work, specifically recovering a description by Bastian [1] when the case $m=1$ is considered.

Figures (5)

• Figure 1: Dynkin diagram of type $\mathbb{A}_n$
• Figure 2: Dynkin diagram of type $\widetilde{\mathbb{A}}_n$
• Figure 3: The quiver $\widetilde{A}_{p,q}$
• Figure 4: The 2-colored mutation class of type $\widetilde{\mathbb{A}}_{3,1}$.
• Figure 5: Example of a quiver in $\mathcal{Q}^m_{p,q}$.

Theorems & Definitions (35)

• Definition 1.1
• Remark 2.1
• Example 2.2
• Definition 2.3
• Proposition 2.4
• Definition 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Lemma 3.4