On the mutation loops of valued quivers

Abstract

A mutation loop of a valued quiver, \(Q\), is a combination of quiver automorphisms and mutations that sends \(Q\)  to itself. Moreover, it will be called symmetric if it sends \(Q\) to \(\epsilon\sigma(Q)\), \(\epsilon \in \{-1, 1\}\) for some permutation \(\sigma\). A global mutation loop of \(Q\) is a mutation loop that is symmetric for every quiver in the mutation class of \(Q\). This class of relations contains all the relations of the global mutations group yield from the group action on the mutation class of \(Q\). We identify which quivers have global mutation loops and provide some of them for each case.