A mutation invariant for skew-symmetrizable matrices

TL;DR

This paper extends a mutation invariant from skew-symmetric to skew-symmetrizable matrices, aiding in classifying matrices in cluster algebra theory, especially when the skew-symmetrizer components are pairwise coprime.

Contribution

It introduces two new extensions of Casals' mutation invariant applicable to skew-symmetrizable matrices with coprime skew-symmetrizers.

Findings

Extended invariant for skew-symmetrizable matrices.

Applicable when skew-symmetrizer components are pairwise coprime.

Enhances classification of mutation classes.

Abstract

Matrix mutation of skew-symmetrizable matrices is foundational in cluster algebra theory. Effective mutation invariants are essential for determining whether two matrices lie in the same mutation class. Casals~\cite{Casals} introduced a binary mutation invariant for skew-symmetric matrices. In this paper, we extend Casals' construction to the skew-symmetrizable setting. When the skew-symmetrizer $d_1,\dots, d_n$ is pairwise coprime, we obtain two distinct extensions of this invariant.