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

TL;DR

This paper provides a detailed combinatorial description of the mutation classes of $m$-colored quivers of type $ ilde{A}_n$, extending previous work and generalizing known cases.

Contribution

It offers an explicit combinatorial framework for understanding $m$-colored mutation classes of $ ilde{A}_n$-quivers, generalizing earlier results for the uncolored case.

Findings

Explicit combinatorial description of $m$-colored mutation classes.

Generalization of Bastian's description for $m=1$ case.

Extension of mutation class understanding to colored quivers.

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.