Lattice Properties of Oriented Exchange Graphs and Torsion Classes

TL;DR

This paper explores the lattice structure of oriented exchange graphs related to quivers with potentials, showing they are semidistributive lattices and connecting them to biclosed subcategories, with implications for maximal green sequences.

Contribution

It proves that lattices of torsion classes are semidistributive and establishes that oriented exchange graphs are semidistributive lattices, extending known structures to new classes of quivers.

Findings

Oriented exchange graphs with finitely many elements are semidistributive lattices.

For type A Dynkin quivers or oriented cycles, these graphs are lattice quotients of biclosed subcategory lattices.

Results address a conjecture on lengths of maximal green sequences.

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 Br\"ustle 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 Br\"ustle, Dupont, and P\'erotin on the lengths of maximal green sequences.