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Algebras and Representation Theory

, Volume 22, Issue 1, pp 43–78 | Cite as

Lattice Properties of Oriented Exchange Graphs and Torsion Classes

  • Alexander GarverEmail author
  • Thomas McConville
Article

Abstract

The exchange graph of a 2-acyclic quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. When the quiver admits a nondegenerate Jacobi-finite potential, the exchange graph admits a natural acyclic orientation called the oriented exchange graph, as shown by Brüstle and Yang. The oriented exchange graph is isomorphic to the Hasse diagram of the poset of functorially finite torsion classes of a certain finite dimensional algebra. We prove that lattices of torsion classes are semidistributive lattices, and we use this result to conclude that oriented exchange graphs with finitely many elements are semidistributive lattices. Furthermore, if the quiver is mutation-equivalent to a type A Dynkin quiver or is an oriented cycle, then the oriented exchange graph is a lattice quotient of a lattice of biclosed subcategories of modules over the cluster-tilted algebra, generalizing Reading’s Cambrian lattices in type A. We also apply our results to address a conjecture of Brüstle, Dupont, and Pérotin on the lengths of maximal green sequences.

Keywords

Torsion class Exchange graph Quiver mutation Lattice 

Mathematics Subject Classification 2010

16G20 18E40 06A07 05E10 

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Notes

Acknowledgements

Alexander Garver thanks Cihan Bahran, Gregg Musiker, Rebecca Patrias, and Hugh Thomas for several helpful conversations. The authors also thank an anonymous referee for carefully commenting on the manuscript.

References

  1. 1.
    Assem, I., Simson, D., Skowroṅski, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1, Volume 65 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (2006). Techniques of representation theoryCrossRefGoogle Scholar
  2. 2.
    Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brüstle, T., Dupont, G., Pérotin, M.: On maximal green sequences. Int. Math. Res. Not. IMRN 2014(16), 4547–4586 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brüstle, T., Yang, D.: Ordered exchange graphs. Adv. Represent. Theory Algebras (ICRA Bielefeld 2012) (2013)Google Scholar
  5. 5.
    Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buan, A.B., Marsh, R., Reiten, I.: Cluster-tilted algebras of finite representation type. J. Algebra 306(2), 412–431 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buan, A.B., Marsh, R., Reiten, I.: Cluster-tilted algebras. Math. Trans. Amer. Soc. 359(1), 323–332 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Buan, A.B., Vatne, D.F.: Derived equivalence classification for cluster-tilted algebras of type A n. J. Algebra 319(7), 2723–2738 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster tilted algebras. Algebr. Represent. Theory 9, 359–376 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (A n case). Trans. Amer. Math. Soc. 358(3), 1347–1364 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Canacki, I., Schroll, S.: Extensions in Jacobian algebras and cluster categories of marked surfaces. Adv. Math. 313, 1–49 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chavez, A.N.: c-vectors and dimension vectors for cluster-finite quivers. Bull. Lond. Math. Soc. (2013)Google Scholar
  13. 13.
    Day, A.: Doubling constructions in lattice theory. Canad. J. Math 44(2), 252–269 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Day, A.: Congruence normality: The characterization of the doubling class of convex sets. Algebra Universalis 31(3), 397–406 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Demonet, L., Iyama, O., Jasso, G.: τ-tilting finite algebras, bricks and g-vectors. Int. Math. Res. Not. IMRN 2017(00), 1–41 (2017)Google Scholar
  16. 16.
    Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations I: Mutations. Selecta Math. (N.S.) 14(1), 59–119 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations II: Applications to cluster algebras. J. Amer. Math. Soc. 23(3), 749–790 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Freese, R., Jezek, J., Nation, J.B.: Free lattices mathematical surveys and monographs, vol. 42. Amer. Math. Soc., Providence (1995)zbMATHGoogle Scholar
  19. 19.
    Grätzer, G., Wehrung, F.: Lattice Theory: Special Topics and Applications, vol. 2. Springer (2016)Google Scholar
  20. 20.
    Iyama, O., Reiten, I., Thomas, H., Todorov, G.: Lattice structure of torsion classes for path algebras. Bull. Lond. Math. Soc. 47(4), 639–650 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kase, R.: Remarks on lengths of maximal green sequences for quivers of type \(\widetilde {A}_{n,1}\). arXiv:1507.02852 (2015)
  22. 22.
    Keller, B.: Cluster algebras and derived categories, pp. 123–183. Eur. Math. Soc., Zürich (2012)zbMATHGoogle Scholar
  23. 23.
    Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations. arXiv:0811.2435 (2008)
  24. 24.
    McConville, T.: Lattice structure of Grid-Tamari orders. J. Combin. Theory Ser. A 148, 27–56 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Reading, N.: Lattice and order properties of the poset of regions in a hyperplane arrangement. Algebra Universalis 50(2), 179–205 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Reading, N.: Cambrian lattices. Adv. Math. 2(205), 313–353 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Thomas, H.: Personal communicationGoogle Scholar
  28. 28.
    Wald, B., Waschbusch, J.: Tame biserial algebras. J. Algebra 95(2), 480–500 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Laboratoire de Combinatoire et d’Informatique MathématiqueUniversité du Québec à MontréalMontréalCanada
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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