# A note on species realizations and nondegeneracy of potentials

## TL;DR

This paper develops a mutation theory for species with potential, demonstrating that certain skew-symmetrizable matrices can have realizations with non-degenerate potentials, partially answering open questions in the field.

## Contribution

It introduces a mutation framework for species with potential and provides examples of matrices with realizations admitting non-degenerate potentials, even when not globally unfoldable.

## Key findings

- A mutation theory for species with potential is established.
- Certain skew-symmetrizable matrices admit non-degenerate potentials.
- Examples include matrices not globally unfoldable but still having non-degenerate potentials.

## Abstract

In this note we show that a mutation theory of species with potential can be defined so that a certain class of skew-symmetrizable integer matrices have a species realization admitting a non-degenerate potential. This gives a partial affirmative answer to a question raised by Jan Geuenich and Daniel Labardini-Fragoso. We also provide an example of a class of skew-symmetrizable $4 \times 4$ integer matrices, which are not globally unfoldable nor strongly primitive, and that have a species realization admitting a non-degenerate potential.