Uniform column sign-coherence and the existence of maximal green sequences

Abstract

In this paper, we prove that each matrix in $$M_{m\times n}({\mathbb {Z}}_{\ge 0})$$ is uniformly column sign-coherent (Definition 2.2 (ii)) with respect to any $$n\times n$$ skew-symmetrizable integer matrix (Corollary 3.3 (ii)). Using such matrices, we introduce the definition of irreducible skew-symmetrizable matrix (Definition 4.1). Based on this, the existence of maximal green sequences for skew-symmetrizable matrices is reduced to the existence of maximal green sequences for irreducible skew-symmetrizable matrices.