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

TL;DR

This paper proves a uniform column sign-coherence property for matrices related to skew-symmetrizable matrices and reduces the problem of finding maximal green sequences to irreducible cases.

Contribution

It introduces the concept of irreducible skew-symmetrizable matrices and links their maximal green sequences to the general case.

Findings

Proves uniform column sign-coherence for a class of matrices.

Defines irreducible skew-symmetrizable matrices.

Reduces maximal green sequence existence to irreducible matrices.

Abstract

In this paper, we prove that each matrix in $M_{m\times n}(\mathbb Z_{\geq0})$ is uniformly column sign-coherent with respect to any $n\times n$ skew-symmetrizable integer matrix. Using such matrices, we introduce the definition of irreducible skew-symmatrizable matrix. Based on this, the existence of a maximal green sequence for a skew-symmetrizable matrices is reduced to the existence of a maximal green sequence for irreducible skew-symmetrizable matrices.