On The Mutations Loops of Valued Quivers

Abstract

A mutation loop of a valued quiver $Q$, is a combination of quiver automorphisms (permutations of vertices and valuations) and mutations that sends $Q$ to itself. In this article we study what we called \emph{global mutations loops} which are sequences formed of mutations only that send each quiver $Q'$ in the mutation class $[Q]$ to $\sigma (Q')$ for some permutation $\sigma$. Based on the weight of $[Q]$ and the shape of $Q$, we identify which quivers have mutations global loops and we provide them for each case.