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


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.


Torsion class Exchange graph Quiver mutation Lattice 

Mathematics Subject Classification 2010

16G20 18E40 06A07 05E10 


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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.


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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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