The Decomposition Algorithm of Skew-symmetrizable Exchange Matrices

TL;DR

This paper presents a combinatorial algorithm to identify skew-symmetrizable matrices associated with surface triangulations and to determine their finite mutation type, extending previous methods.

Contribution

It introduces a generalized algorithm for classifying skew-symmetrizable matrices and assessing their mutation finiteness, broadening the scope of prior approaches.

Findings

Algorithm successfully classifies matrices related to surface triangulations.

Determines whether a skew-symmetrizable matrix has finite mutation type.

Extends previous algorithms to a more general class of matrices.

Abstract

Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admits unfoldings to skew-symmetric matrices. We develop an combinatorial algorithm that determines if a given skew-symmetrizable matrix is of such type. This algorithm generalizes the one in \cite{WG}. As a corollary, we use this algorithm to determine if a given skew-symmetrizable matrix has finite mutation type.