The Decomposition Algorithm for Skew-Symmetrizable Exchange Matrices

Abstract

Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admit unfoldings to skew-symmetric matrices. We develop a combinatorial algorithm that determines if a given skew-symmetrizable matrix is of such type. This algorithm generalizes the one in Weiwen Gu's Decomposition Algorithm for Median Graph of Triangulation of a Bordered 2D Surface. As a corollary, we use this algorithm to determine if a given skew-symmetrizable matrix has finite mutation type.