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

  • Peigen Cao
  • Fang LiEmail author


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.


Cluster algebra Sign-coherence Maximal green sequence Green-to-red sequence 

Mathematics Subject Classification

13F60 05E40 



This project is supported by the National Natural Science Foundation of China (Nos. 11671350 and 11571173) and the Zhejiang Provincial Natural Science Foundation of China (No. LY18A010032).


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang University (Yuquan Campus)HangzhouPeople’s Republic of China

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