A mutation invariant for skew-symmetrizable matrices

TL;DR

The paper extends the binary δ-invariant, originally for skew-symmetric matrices, to the broader class of skew-symmetrizable integer matrices by using the symmetrization $\mathfrak{S}(B)$ and defining $\delta(B)=\det(\mathfrak{S}(B))\bmod 4$. It proves that mutation does not change $\delta$ modulo $4$ (and, under the pairwise coprime skew-symmetrizer condition, yields a second invariant $\delta'$ via an alternate symmetrization $\mathfrak{S}'(B)$ modulo $4\prod d_i$). The approach hinges on establishing integrality of $\det(\mathfrak{S}(B))$, showing permutation invariance of $\delta$ modulo $4$, and analyzing mutation via rank-two perturbations together with careful congruence arguments. These invariants provide a practical tool for distinguishing mutation-classes in cluster algebra theory and extend Casals' framework to a wider class of matrices.

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.

Theorems & Definitions (19)

• Definition 1.1
• Lemma 1.2
• Definition 1.3
• Lemma 1.4
• Theorem 1.1
• Remark 1.5
• Example 1.6
• Lemma 2.1
• proof
• proof : Proof of Lemma \ref{['lem:integer']}