Lattice Properties of Oriented Exchange Graphs and Torsion Classes
Author(s)
Garver, Alexander
McConville, Thomas
Date Issued
December 2017
Journal
Algebras and Representation Theory
Publisher
Springer Science and Business Media LLC
Citation
Garver, Alexander and Thomas McConville. “Lattice Properties of Oriented Exchange Graphs and Torsion Classes.” Algebras and Representation Theory 22, 1 (December 2017): 43–78. © 2017 Springer Science Business Media B.V., part of Springer Nature
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
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DOI of Published Version
https://doi.org/10.1007/s10468-017-9757-1
