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Coxeter group
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{{Short description|Group that admits a formal description in terms of reflections}} In [[mathematics]], a '''Coxeter group''', named after [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]], is an [[group (mathematics)|abstract group]] that admits a [[Presentation of a group|formal description]] in terms of [[Reflection (mathematics)|reflections]] (or [[Kaleidoscope|kaleidoscopic mirrors]]). Indeed, the finite Coxeter groups are precisely the finite Euclidean [[reflection group]]s; for example, the [[symmetry group]] of each [[regular polyhedron]] is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of [[Symmetry in mathematics|symmetries]] and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups,<ref name="Coxeter1934">{{cite journal|title=Discrete groups generated by reflections|last=Coxeter|first=H. S. M.|journal=Annals of Mathematics|volume=35|pages=588โ621 |year=1934|issue=3 |citeseerx=10.1.1.128.471|doi=10.2307/1968753|jstor=1968753|language=en}}</ref> and finite Coxeter groups were classified in 1935.<ref name="Coxeter1935">{{cite journal |title=The complete enumeration of finite groups of the form <math>r_{i}^{2}=(r_{i}r_{j})^{k_{ij}}=1</math>|last=Coxeter|first=H. S. M. |journal=Journal of the London Mathematical Society |pages=21โ25 |date=January 1935|language=en|doi=10.1112/jlms/s1-10.37.21}}</ref> Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of [[regular polytope]]s, and the [[Weyl group]]s of [[simple Lie algebra]]s. Examples of infinite Coxeter groups include the [[triangle group]]s corresponding to [[Tessellation#Overview|regular tessellation]]s of the [[Euclidean plane]] and the [[Hyperbolic space|hyperbolic plane]], and the Weyl groups of infinite-dimensional [[KacโMoody algebra]]s.<ref>{{cite book|title=Lie Groups and Lie Algebras|last=Bourbaki|first=Nicolas|year=2002|chapter=4-6|publisher=Springer|language=en|isbn=978-3-540-42650-9|zbl=0983.17001|series=Elements of Mathematics}}</ref><ref>{{cite book|title=Reflection Groups and Coxeter Groups|last=Humphreys|first=James E.|url=https://sites.math.washington.edu/~billey/classes/reflection.groups/references/Humphreys.ReflectionGroupsAndCoxeterGroups.pdf|series=Cambridge Studies in Advanced Mathematics|volume=29|access-date=2023-11-18|year=1990|publisher=Cambridge University Press|doi=10.1017/CBO9780511623646|isbn=978-0-521-43613-7|zbl=0725.20028}}</ref><ref>{{cite book|title=The Geometry and Topology of Coxeter Groups|last=Davis|first=Michael W.|url=https://people.math.osu.edu/davis.12/davisbook.pdf|access-date=2023-11-18|year=2007|publisher=Princeton University Press|language=en|isbn=978-0-691-13138-2|zbl=1142.20020}}</ref>
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