Mutation Of Matrices Over Group Rings

TL;DR

This paper introduces a detailed definition of mutation for skew symmetrizable matrices over group rings, connecting it to quiver folding and mutation, including cases with non-zero diagonal entries, through a generalized mutation concept.

Contribution

It provides a new mutation rule for matrices over group rings, extending existing frameworks to include matrices with non-zero diagonals and their relation to quiver mutations.

Findings

Defined mutation for matrices over group rings

Connected matrix mutation to quiver folding and mutation

Extended mutation rules to matrices with non-zero diagonals

Abstract

We give a precise definition of mutation of skew symmetrizable matrices over group rings and relate it to folding and mutation of quivers with symmetries. These matrices can have non-zero diagonal entries and we explain a mutation rule in some of these cases as well. This new rule comes from a notion of a generalized mutation of an entire quiver or sub-quiver.