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Reflection subgroups of Coxeter systems. (English) Zbl 0712.20026
Let \((W,R)\) be a Coxeter system, \(T\) the set of reflections, \(W'=\langle W'\cap T\rangle\) a reflection subgroup. The author shows that a certain subset of \(T\) is a set of Coxeter generators of \(W'\). He also gives a geometric criterion for a set of reflections to be the set of canonical generators of a reflection subgroup and classifies the isomorphism types of reflection subgroups of affine Weyl groups.
Reviewer: E.W.Ellers

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20H15 Other geometric groups, including crystallographic groups
20F05 Generators, relations, and presentations of groups
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[1] Bourbaki, N, Groupes et algèbres de Lie, (1968), Hermann Paris, Chaps. 4-6
[2] Carter, R, Conjugacy classes in the Weyl group, Compositio math., 25, 1-59, (1972) · Zbl 0254.17005
[3] Coxeter, H.S.M, Finite groups generated by reflections and their subgroups generated by reflections, (), 466-482 · Zbl 0010.15403
[4] Deodhar, V, On the root system of a Coxeter group, Comm. algebra, 10, 611-630, (1982) · Zbl 0491.20032
[5] Deodhar, V, Some characterizations of Coxeter groups, Enseign. math., 32, 2, 111-120, (1986) · Zbl 0611.20030
[6] {\scV. Deodhar}, A note on subgroups generated by reflections in Coxeter groups, preprint. · Zbl 0688.20028
[7] Dyer, M, Hecke algebras and reflection in Coxeter groups, ()
[8] Kac, V, Infinite dimensional Lie algebras, (1985), Cambridge Univ. Press Cambridge
[9] Matsumoto, H, Générateurs et relations des groupes de Weyl généralisés, Acad. sci. Paris, 258, 3419-3422, (1964) · Zbl 0128.25202
[10] Springer, T, Some remarks on involutions in Coxeter groups, Comm. algebra, 10, 631-636, (1982) · Zbl 0531.20016
[11] Steinberg, R, Endomorphisms of linear algebraic groups, Mem. amer. math. soc., 80, (1968) · Zbl 0164.02902
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