Strongly primitive species with potentials I: Mutations

TL;DR

This paper develops a mutation theory for species with potentials derived from skew-symmetrizable matrices, expanding the framework of quiver mutations to include matrices without global unfoldings, thus broadening the scope of cluster algebra models.

Contribution

It introduces a mutation theory for species with potentials associated with a new class of skew-symmetrizable matrices that lack global unfoldings.

Findings

Defines mutation operations for species with potentials from skew-symmetrizable matrices.

Includes matrices not admitting global unfoldings in the mutation framework.

Extends mutation theory to a broader class of matrices in cluster algebra context.

Abstract

Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.