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Lattice Properties of Oriented Exchange Graphs and Torsion Classes

Authors
  • Garver, Alexander1
  • McConville, Thomas2
  • 1 Université du Québec à Montréal, Laboratoire de Combinatoire et d’Informatique Mathématique, Montréal, QC H3C 3P8, Canada , Montréal (Canada)
  • 2 Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA , Cambridge (United States)
Type
Published Article
Journal
Algebras and Representation Theory
Publisher
Springer Netherlands
Publication Date
Dec 18, 2017
Volume
22
Issue
1
Pages
43–78
Identifiers
DOI: 10.1007/s10468-017-9757-1
Source
Springer Nature
Keywords
License
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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.

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