On The Mutations Loops of Valued Quivers

TL;DR

This paper investigates mutation loops in valued quivers, identifying conditions under which sequences of mutations return quivers to their original form up to permutation, thus deepening understanding of quiver mutation classes.

Contribution

It introduces the concept of global mutation loops in valued quivers and characterizes which quivers admit such loops based on their weights and shape.

Findings

Identifies conditions for the existence of global mutation loops.

Provides explicit examples of quivers with mutation loops.

Classifies quivers based on their mutation properties.

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.