Reflection subgroups of finite and affine Weyl groups

Dyer, M.; Lehrer, G.

doi:10.1090/S0002-9947-2011-05298-0

Reflection subgroups of finite and affine Weyl groups
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by M. J. Dyer and G. I. Lehrer PDF

Trans. Amer. Math. Soc. 363 (2011), 5971-6005 Request permission

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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