Weighted quivers

TL;DR

This paper extends quiver mutation theory to weighted quivers with group-valued weights, providing new proofs of known properties and classifying weights on tame quivers.

Contribution

It introduces weights into quiver mutation, proving that mutation works with weights under certain conditions and classifies weights on tame quivers.

Findings

Mutation of weighted quivers is valid when cycle weights have trivial product.

Provides a new proof of sign coherence of c-vectors.

Classifies all weights on tame quivers.

Abstract

A "weight" on a quiver $Q$ with values in a group $G$ is a function which assigns an element of $G$ for each arrow in $Q$. This paper shows that the essential steps in the mutation of quivers with potential [DWZ] goes through with weights provided that the weights on each cycle in the potential have trivial product. This gives another proof of the sign coherence of $c$-vectors. We also classify all weights on tame quivers.