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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Reflection subgroups of finite and affine Weyl groups


Authors: M. J. Dyer and G. I. Lehrer
Journal: Trans. Amer. Math. Soc. 363 (2011), 5971-6005
MSC (2000): Primary 20F55, 51F15
Published electronically: May 2, 2011
MathSciNet review: 2817417
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Abstract: We give complete classifications of the reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case-free proof is given of the well-known classification of the isomorphism classes of reflection subgroups using completed Dynkin diagrams, for which there seems to be no convenient source in the literature. This is used as a basis for treating the affine case, where we give two distinct `on the nose' classifications of reflection subgroups, as well as reproving Coxeter's conjecture on the isomorphism classes of reflection groups which occur. Geometric and combinatorial aspects of root systems play an essential role. Parameter sets for various types of subsets of roots are interpreted in terms of alcove geometry and the Tits cone, and combinatorial identities are derived.


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Additional Information

M. J. Dyer
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, 2006 Australia
Address at time of publication: Department of Mathematics, 25 Hurley Building, University of Notre Dame, Notre Dame, Indiana 46556

G. I. Lehrer
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney, 2006 Australia

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05298-0
PII: S 0002-9947(2011)05298-0
Received by editor(s): December 13, 2009
Received by editor(s) in revised form: January 26, 2010
Published electronically: May 2, 2011
Article copyright: © Copyright 2011 American Mathematical Society