Abstract
We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type \(\mathbb {A}\). We prove that such sequences have length \(n+t\), where n is the number of vertices and t is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Adachi, T., Iyama, O., Reiten, I.: \(\tau \)-tilting theory. Compos. Math. 150(3), 415–452 (2014)
Alim, M., Cecotti, S., Córdova, C., Espahbodi, S., Rastogi, A., Vafa, C.: BPS quivers and spectra of complete N= 2 quantum field theories. Commun. Math. Phys. 323, 1185–1227 (2013)
Brüstle, T., Dupont, G., Perotin, M.: On maximal green sequences. Int. Math. Res. 2014(16), 4547–4586 (2014)
Brüstle, T., Hermes, S., Igusa, K., Todorov, G.: Semi-invariant pictures and two conjectures on maximal green sequences (2015). arXiv:1503.07945
Brüstle, T., Qiu, Y.: Tagged mapping class groups: Auslander–Reiten translation. Math. Z. 279(3–4), 1103–1120 (2015)
Brüstle, T., Yang, D.: Ordered exchange graphs (2014). arXiv:1302.6045
Bucher, E.: Maximal green sequences for cluster algebras associated to the n-torus (2014). arXiv:1412.3713
Bucher, E., Mills, M.: Maximal green sequences for cluster algebras associated to the orientable surfaces of genus n with arbitrary punctures (2015). arXiv:1503.06207
Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (\(A_n\) case). Trans. Am. Math. Soc. 358(3), 1347–1364 (2006)
Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations II: applications to cluster algebras. J. Am. Math. Soc. 23(3), 749–790 (2010)
Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008)
Fomin, S., Thurston, D.: Cluster algebras and triangulated surfaces. Part II: lambda lengths (2012). arXiv:1210.5569
Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15, 497–529 (2002)
Fomin, S., Zelevinsky, A.: Cluster algebras II: finite type classification. Invent. Math. 154, 63–121 (2003)
Garver, A., McConville, T.: Lattice properties of oriented exchange graphs and torsion classes (2015). arXiv:1507.04268
Garver, A., Musiker, G.: On maximal green sequences for type A quivers (2014). arXiv:1403.6149
Kase, R.: Remarks on lengths of maximal green sequences for quivers of type \(\tilde{A}_{n,1}\) (2015). arXiv:1507.02852
Keller, B.: On cluster theory and quantum dilogarithm identities. In: Skowronski, A., Yamagata, K. (eds.) Representations of Algebras and Related Topics, pp. 85–116. EMS Series of Congress Reports, European Mathematical Society, Helsinki (2011)
Ladkrani, S.: On cluster algebras from once punctured closed surfaces (2013). arXiv:1310.4454
Mozgovoy, S., Reineke, M.: On the noncomutative Donaldson–Thomas invariants arising from brane tiltings. Adv. Math. 223, 1521–1544 (2010)
Muller, G.: The existence of a maximal green sequence is not invariant under quiver mutation (2015). arXiv:1503.04675
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was carried out at the University of Connecticut 2015 math REU funded by National Science Foundation under DMS-1262929. The fourth author was also supported by the National Science Foundation CAREER Grant DMS-1254567.
Rights and permissions
About this article
Cite this article
Cormier, E., Dillery, P., Resh, J. et al. Minimal length maximal green sequences and triangulations of polygons. J Algebr Comb 44, 905–930 (2016). https://doi.org/10.1007/s10801-016-0694-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-016-0694-6