The coloured mutation class of ${\mathbb{A}}_n$ quivers

Abstract

In this paper we give an explicit and pure combinatorial description of the $m$-coloured quivers that appears in the $m$-coloured mutation class of a quiver of type ${\mathbb{A}}_n$. The $m$-coloured mutation defined by Buan and Thomas in \cite{BT} generalizes the well-known quiver mutation of Fomin and Zelevinsky \cite{FZ}. In particular, our description generalizes a result of Buan and Vatne, \cite{BV}, which we recover when $m=1$.