Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite type

Nasr-Isfahani, Alireza

doi:10.1090/proc/14523

Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite type
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by Alireza Nasr-Isfahani PDF

Proc. Amer. Math. Soc. 147 (2019), 2775-2782 Request permission

Abstract:

Let $\Lambda$ be a cluster-tilted algebra of finite type over an algebraically closed field and let $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered from right to left in the Auslander-Reiten quiver of $\Lambda$ form a maximal forward hom-orthogonal sequence of $\Lambda$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $\Lambda$. Thus we give a proof of Igusa-Todorov’s conjecture.

References

Murad Alim, Sergio Cecotti, Clay Córdova, Sam Espahbodi, Ashwin Rastogi, and Cumrun Vafa, $\scr {N}=2$ quantum field theories and their BPS quivers, Adv. Theor. Math. Phys. 18 (2014), no. 1, 27–127. MR 3268234
I. Assem, T. Brüstle, and R. Schiffler, Cluster-tilted algebras as trivial extensions, Bull. Lond. Math. Soc. 40 (2008), no. 1, 151–162. MR 2409188, DOI 10.1112/blms/bdm107
Ibrahim Assem, Thomas Brüstle, and Ralf Schiffler, On the Galois coverings of a cluster-tilted algebra, J. Pure Appl. Algebra 213 (2009), no. 7, 1450–1463. MR 2497589, DOI 10.1016/j.jpaa.2008.12.008
Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR 2197389, DOI 10.1017/CBO9780511614309
Karin Baur and Alireza Nasr-Isfahani, Strongness of companion bases for cluster-tilted algebras of finite type, Proc. Amer. Math. Soc. 146 (2018), no. 6, 2409–2416. MR 3778144, DOI 10.1090/proc/13977
Thomas Brüstle, Grégoire Dupont, and Matthieu Pérotin, On maximal green sequences, Int. Math. Res. Not. IMRN 16 (2014), 4547–4586. MR 3250044, DOI 10.1093/imrn/rnt075
Aslak Bakke Buan, Robert Marsh, Markus Reineke, Idun Reiten, and Gordana Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), no. 2, 572–618. MR 2249625, DOI 10.1016/j.aim.2005.06.003
Aslak Bakke Buan, Robert J. Marsh, and Idun Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359 (2007), no. 1, 323–332. MR 2247893, DOI 10.1090/S0002-9947-06-03879-7
Aslak Bakke Buan, Bethany R. Marsh, and Idun Reiten, Cluster mutation via quiver representations, Comment. Math. Helv. 83 (2008), no. 1, 143–177. MR 2365411, DOI 10.4171/CMH/121
Philippe Caldero, Frédéric Chapoton, and Ralf Schiffler, Quivers with relations and cluster tilted algebras, Algebr. Represent. Theory 9 (2006), no. 4, 359–376. MR 2250652, DOI 10.1007/s10468-006-9018-1
S. Cecotti, C. Córdova, and C. Vafa, Braids, walls and mirrors, arXiv:1110.2115v1 [hep-th], (2011).
Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), no. 1, 59–119. MR 2480710, DOI 10.1007/s00029-008-0057-9
Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), no. 3, 749–790. MR 2629987, DOI 10.1090/S0894-0347-10-00662-4
Piotr Dowbor and Andrzej Skowroński, On Galois coverings of tame algebras, Arch. Math. (Basel) 44 (1985), no. 6, 522–529. MR 797444, DOI 10.1007/BF01193992
Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, DOI 10.1090/S0894-0347-01-00385-X
Alexander Garver, Thomas McConville, and Khrystyna Serhiyenko, Minimal length maximal green sequences, Adv. in Appl. Math. 96 (2018), 76–138. MR 3767505, DOI 10.1016/j.aam.2017.12.008
K. Igusa, Maximal green sequences for cluster-tilted algebras of finite type, arXiv:1706.06503v1 [math.RT], (2017).
Bernhard Keller, On cluster theory and quantum dilogarithm identities, Representations of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 85–116. MR 2931896, DOI 10.4171/101-1/3
Bin Zhu, Equivalences between cluster categories, J. Algebra 304 (2006), no. 2, 832–850. MR 2264281, DOI 10.1016/j.jalgebra.2006.03.012