The coloured mutation class of $\mathbb{D}_n$- quivers and their application to $m$-cluster tilted algebras

TL;DR

This paper provides a combinatorial description of the $m$-coloured mutation class of $ ext{D}_n$ quivers and explores their application to $m$-cluster tilted algebras, extending previous results.

Contribution

It offers an explicit combinatorial characterization of $m$-coloured quivers in the mutation class of type $ ext{D}_n$, generalizing known results for $m=1$.

Findings

Characterization of $m$-coloured mutation class of $ ext{D}_n$ quivers.

Description of Gabriel quivers for $m$-cluster-tilted algebras of type $ ext{D}_n$.

Extension of Vatne's result to general $m$.

Abstract

In this paper, we present an explicit and purely combinatorial characterization of the $m$-coloured quivers that appear within the $m$-coloured mutation class of a quiver of type $\mathbb{D}_n$. The $m$-coloured mutation, as defined by Buan and Thomas in \cite{BT}, generalises the well-known quiver mutation introduced by Fomin and Zelevinsky \cite{FZ}. Consequently, we derive a comprehensive description of the Gabriel quivers associated with $m$-cluster-tilted algebras of type $\mathbb{D}_n$. Notably, our characterization extends a result by Vatne, \cite{Va}, which we recover when $m=1$.