Lattice Properties of Oriented Exchange Graphs and Torsion Classes
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 an...
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| Language: | English |
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Springer Science and Business Media LLC
2021
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| Online Access: | https://hdl.handle.net/1721.1/130080 |
| _version_ | 1811096343633985536 |
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| author | Garver, Alexander McConville, Thomas |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Garver, Alexander McConville, Thomas |
| author_sort | Garver, Alexander |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T16:42:20Z |
| format | Article |
| id | mit-1721.1/130080 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:42:20Z |
| publishDate | 2021 |
| publisher | Springer Science and Business Media LLC |
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| spelling | mit-1721.1/1300802022-09-29T20:56:36Z Lattice Properties of Oriented Exchange Graphs and Torsion Classes Garver, Alexander McConville, Thomas Massachusetts Institute of Technology. Department of Mathematics 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. 2021-03-04T15:43:58Z 2021-03-04T15:43:58Z 2017-12 2017-04 2019-02-07T05:19:06Z Article http://purl.org/eprint/type/JournalArticle 1386-923X 1572-9079 https://hdl.handle.net/1721.1/130080 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 en https://doi.org/10.1007/s10468-017-9757-1 Algebras and Representation Theory Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media B.V., part of Springer Nature text/xml application/pdf Springer Science and Business Media LLC Springer Netherlands |
| spellingShingle | Garver, Alexander McConville, Thomas Lattice Properties of Oriented Exchange Graphs and Torsion Classes |
| title | Lattice Properties of Oriented Exchange Graphs and Torsion Classes |
| title_full | Lattice Properties of Oriented Exchange Graphs and Torsion Classes |
| title_fullStr | Lattice Properties of Oriented Exchange Graphs and Torsion Classes |
| title_full_unstemmed | Lattice Properties of Oriented Exchange Graphs and Torsion Classes |
| title_short | Lattice Properties of Oriented Exchange Graphs and Torsion Classes |
| title_sort | lattice properties of oriented exchange graphs and torsion classes |
| url | https://hdl.handle.net/1721.1/130080 |
| work_keys_str_mv | AT garveralexander latticepropertiesoforientedexchangegraphsandtorsionclasses AT mcconvillethomas latticepropertiesoforientedexchangegraphsandtorsionclasses |