Reflection group presentations arising from cluster algebras
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- by Michael Barot and Bethany R. Marsh PDF
- Trans. Amer. Math. Soc. 367 (2015), 1945-1967 Request permission
Abstract:
We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.References
- Michael Barot, Christof Geiss, and Andrei Zelevinsky, Cluster algebras of finite type and positive symmetrizable matrices, J. London Math. Soc. (2) 73 (2006), no. 3, 545–564. MR 2241966, DOI 10.1112/S0024610706022769
- M. Barot and D. Rivera, Generalized Serre relations for Lie algebras associated with positive unit forms, J. Pure Appl. Algebra 211 (2007), no. 2, 360–373. MR 2340453, DOI 10.1016/j.jpaa.2007.01.008
- 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
- P. Caldero, F. Chapoton, and R. Schiffler, Quivers with relations arising from clusters ($A_n$ case), Trans. Amer. Math. Soc. 358 (2006), no. 3, 1347–1364. MR 2187656, DOI 10.1090/S0002-9947-05-03753-0
- P. J. Cameron, J. J. Seidel, and S. V. Tsaranov, Signed graphs, root lattices, and Coxeter groups, J. Algebra 164 (1994), no. 1, 173–209. MR 1268332, DOI 10.1006/jabr.1994.1059
- 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
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. MR 2004457, DOI 10.1007/s00222-003-0302-y
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- M. J. Parsons, On indecomposable modules over cluster-tilted algebras of type $A$, PhD Thesis, University of Leicester, 2007.
- M. J. Parsons, Companion bases for cluster-tilted algebras, Algebr. Represent. Theory 17 (2014), no. 3, 775–808. DOI 10.1007/s10468-013-9418-y.
- Claus Michael Ringel, Cluster-concealed algebras, Adv. Math. 226 (2011), no. 2, 1513–1537. MR 2737792, DOI 10.1016/j.aim.2010.08.014
Additional Information
- Michael Barot
- Affiliation: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México, Distrito Federal, C.P. 04510 México
- Email: barot@matem.unam.mx
- Bethany R. Marsh
- Affiliation: School of Mathematics, University of Leeds, Leeds LS2 9JT, England
- MR Author ID: 614298
- ORCID: 0000-0002-4268-8937
- Received by editor(s): February 15, 2012
- Received by editor(s) in revised form: March 1, 2013
- Published electronically: October 16, 2014
- Additional Notes: This work was supported by DGAPA, Universidad Nacional Autónoma de México, the Engineering and Physical Sciences Research Council [grant number EP/G007497/1] and the Institute for Mathematical Research (FIM, Forschungsinstitut für Mathematik) at the ETH, Zürich
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 1945-1967
- MSC (2010): Primary 13F60, 20F55, 51F15; Secondary 16G20
- DOI: https://doi.org/10.1090/S0002-9947-2014-06147-3
- MathSciNet review: 3286504