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.

This is a preview of subscription content, log in to check access.

Fig. 1

References

  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 

Download references

Acknowledgements

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).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Fang Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

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

Mathematics Subject Classification

  • 13F60
  • 05E40