Skip to main content

Lengths of maximal green sequences for tame path algebras

Abstract

In this paper, we study the maximal length of maximal green sequences for quivers of type \(\widetilde{{\mathbf {D}}}\) and \(\widetilde{{\mathbf {E}}}\) by using the theory of tilting mutation. We show that the maximal length does not depend on the choice of the orientation and determine it explicitly. Moreover, we give a program which counts all maximal green sequences by length for a given Dynkin/extended Dynkin quiver.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Adachi, T., Iyama, O., Reiten, I.: \(\tau \)-tilting theory. Compos. Math. 150(3), 415–452 (2014). https://doi.org/10.1112/S0010437X13007422

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Apruzzese, P.J., Igusa, K.: Stability conditions for affine type A. Algebras Represent. Theory 23(5), 2079–2111 (2020). https://doi.org/10.1007/s10468-019-09926-z

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. London Mathematical Society Student Texts. Techniques of Representation Theory, vol. 65. Cambridge University Press, Cambridge (2006). https://doi.org/10.1017/CBO9780511614309

    Book  MATH  Google Scholar 

  4. 4.

    Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995). https://doi.org/10.1017/CBO9780511623608

    Book  MATH  Google Scholar 

  5. 5.

    Brüstle, T., Dupont, G., Pérotin, M.: On maximal green sequences. Int. Math. Res. Not. IMRN 16, 4547–4586 (2014). https://doi.org/10.1093/imrn/rnt075

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Brüstle, T., Smith, D., Treffinger, H.: Wall and chamber structure for finite-dimensional algebras. Adv. Math. 354, 106746 (2019). https://doi.org/10.1016/j.aim.2019.106746

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Cormier, E., Dillery, P., Resh, J., Serhiyenko, K., Whelan, J.: Minimal length maximal green sequences and triangulations of polygons. J. Algebraic Comb. 44(4), 905–930 (2016). https://doi.org/10.1007/s10801-016-0694-6

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscripta Math. 6, 71–103; correction, ibid. 6 (1972), 309 (1972). https://doi.org/10.1007/BF01298413

  9. 9.

    Garver, A., McConville, T.: Lattice properties of oriented exchange graphs and torsion classes. Algebras Represent. Theory 22(1), 43–78 (2019). https://doi.org/10.1007/s10468-017-9757-1

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Garver, A., McConville, T., Serhiyenko, K.: Minimal length maximal green sequences. Adv. Appl. Math. 96, 76–138 (2018). https://doi.org/10.1016/j.aam.2017.12.008

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Hermes, S., Igusa, K.: The no gap conjecture for tame hereditary algebras. J. Pure Appl. Algebra 223(3), 1040–1053 (2019). https://doi.org/10.1016/j.jpaa.2018.05.013

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Ingalls, C., Thomas, H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533–1562 (2009). https://doi.org/10.1112/S0010437X09004023

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Jasso, G.: Reduction of \(\tau \)-tilting modules and torsion pairs. Int. Math. Res. Not. IMRN 16, 7190–7237 (2015). https://doi.org/10.1093/imrn/rnu163

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Kahn, A.B.: Topological sorting of large networks. Commun. ACM 5(11), 558–562 (1962). https://doi.org/10.1145/368996.369025

    Article  MATH  Google Scholar 

  15. 15.

    Keller, B.: On cluster theory and quantum dilogarithm identities. In: Representations of algebras and related topics, EMS Series of Congress Reports, pp. 85–116. European Mathematical Society, Zürich (2011). https://doi.org/10.4171/101-1/3

  16. 16.

    Keller, B.: Cluster algebras and derived categories. In: Derived categories in algebraic geometry, EMS Series of Congress Reports, pp. 123–183. European Mathematical Society, Zürich (2012)

  17. 17.

    Keller, B., Demonet, L.: A survey on maximal green sequences. In: Representation Theory and Beyond. Contemporary Mathematics, vol. 758, pp. 267–286. American Mathematical Society, Providence, RI (2020). https://doi.org/10.1090/conm/758/15239

  18. 18.

    Ladkani, S.: Universal derived equivalences of posets of cluster tilting objects (2007)

  19. 19.

    Qiu, Y.: \(C\)-sortable words as green mutation sequences. Proc. Lond. Math. Soc. (3) 111(5), 1052–1070 (2015). https://doi.org/10.1112/plms/pdv046

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, Berlin (1984)

    Book  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ryoichi Kase.

Additional information

Publisher's Note

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

R. Kase is supported by KAKENHI Grant Number JP17K14169.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kase, R., Nakashima, K. Lengths of maximal green sequences for tame path algebras. Res Math Sci 8, 59 (2021). https://doi.org/10.1007/s40687-021-00290-3

Download citation

Keywords

  • Maximal green sequence
  • (\(\tau \)-)tilting theory
  • Quiver representation
  • Tame path algebra
  • Support (\(\tau \)-)tilting poset

Mathematics Subject Classification

  • 16G20 Primary
  • 06-04
  • 16G60