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Reflection group presentations arising from cluster algebras


Authors: Michael Barot and Robert J. Marsh
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
Published electronically: October 16, 2014
MathSciNet review: 3286504
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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.


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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

Robert J. Marsh
Affiliation: School of Mathematics, University of Leeds, Leeds LS2 9JT, England
Email: marsh@maths.leeds.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2014-06147-3
Keywords: Reflection group, Weyl group, finite type, Dynkin diagram, cluster algebra, companion basis, quasi-Cartan companion, mutation, diagram, presentation, cycle, Coxeter graph
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.