Lattice Properties of Oriented Exchange Graphs and Torsion Classes

By Alexander Garver, Thomas McConville. 2015

49 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6ab05a57e14468700617b58dbafa5dd80837c4fd

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Rowmotion in slow motion (2017, background): “This partial order (torsion pairs, ordered by inclusion of torsion classes; which can also be described as inclusion order on torsion classes) has been considered in [20, 31, 37, 53].”
A survey on maximal green sequences (2019, methodology): “d and green) sequences of mutations (cf. section 7.6 of [77]) and that this generalized version shows that if there is a reddening sequence, the Jacobi algebra is still ﬁnite-dimensional. We refer to [52] for applications of the theorem to the study of lattice properties of oriented exchange graphs.”