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


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.

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Fig. 1


  1. 1.

    Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: BPS quivers and spectra of complete \(N=2\) quantum field theories. Comm. Math. Phys. 323, 1185–1227 (2013)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Brüstle, T., Dupont, G., Pérotin, M.: On maximal green sequences. Int. Math. Res. Not. 2014(16), 4547–4586 (2014)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Brüstle, T., Hermes, S., Igusa, K., Todorov, G.: Semi-invariant pictures and two conjectures on maximal green sequences. J. Algebra 473, 80–109 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bucher, E., Mills, M.R.: Maximal green sequences for cluster algebras associated with the \(n\)-torus with arbitrary punctures. J. Algebraic Combin. 47(3), 345–356 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Fomin, S., Zelevinsky, A.: Cluster algebras. IV. Coefficients. Compos. Math. 143, 112–164 (2007)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Garver, A., Musiker, G.: On maximal green sequences for type \(A\) quiver. J. Algebraic Combin. 45(2), 553–599 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Gross, M., Hacking, P., Keel, S., Kontsevich, M.: Canonical bases for cluster algebras. J. Amer. Math. Soc. 31, 497–608 (2018)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Keller, B.: On cluster theory and quantum dilogarithm identities. Representations of Algebras and Related Topics, EMS Series OF Congress Report, pp. 85–116. European Mathematical Society, Zürich (2011)

  11. 11.

    Ladkani, S.: On Cluster Algebras from once Punctured Closed Surfaces. arXiv:1310.4454

  12. 12.

    Muller, G.: The existence of a maximal green sequence is not invariant under quiver mutation. Electron. J. Combin. 23(2), 1–23 (2016). (Paper 2.47)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Mills, M.R.: Maximal green sequences for quivers of finite mutation type. Adv. Math. 319, 182–210 (2017)

    MathSciNet  Article  Google Scholar 

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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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Correspondence to Fang Li.

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Cao, P., Li, F. Uniform column sign-coherence and the existence of maximal green sequences. J Algebr Comb 50, 403–417 (2019). https://doi.org/10.1007/s10801-018-0861-z

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

Mathematics Subject Classification

  • 13F60
  • 05E40