Dyer, Matthew Reflection subgroups of Coxeter systems. (English) Zbl 0712.20026 J. Algebra 135, No. 1, 57-73 (1990). 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 Cited in 4 ReviewsCited in 100 Documents MSC: 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20H15 Other geometric groups, including crystallographic groups 20F05 Generators, relations, and presentations of groups Keywords:root systems; cocycles; Coxeter systems; reflections; reflection subgroups; Coxeter generators; affine Weyl groups PDFBibTeX XMLCite \textit{M. Dyer}, J. Algebra 135, No. 1, 57--73 (1990; Zbl 0712.20026) Full Text: DOI References: [1] Bourbaki, N., Groupes et algèbres de Lie (1968), Hermann: 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, (Proc. Cambridge Philos. Soc., 30 (1934)), 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] V. Deodhar, A note on subgroups generated by reflections in Coxeter groups, preprint.; V. 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, (Ph.D. Thesis (1987), University of Sydney) [8] Kac, V., Infinite Dimensional Lie Algebras (1985), Cambridge Univ. Press: 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.