Group that admits a formal description in terms of reflections
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; 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 symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups,[1] and finite Coxeter groups were classified in 1935.[2]
Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.[3][4][5]
^Coxeter, H. S. M. (1934). "Discrete groups generated by reflections". Annals of Mathematics. 35 (3): 588–621. CiteSeerX 10.1.1.128.471. doi:10.2307/1968753. JSTOR 1968753.
^Coxeter, H. S. M. (January 1935). "The complete enumeration of finite groups of the form ". Journal of the London Mathematical Society: 21–25. doi:10.1112/jlms/s1-10.37.21.
^Bourbaki, Nicolas (2002). "4-6". Lie Groups and Lie Algebras. Elements of Mathematics. Springer. ISBN 978-3-540-42650-9. Zbl 0983.17001.
^Humphreys, James E. (1990). Reflection Groups and Coxeter Groups(PDF). Cambridge Studies in Advanced Mathematics. Vol. 29. Cambridge University Press. doi:10.1017/CBO9780511623646. ISBN 978-0-521-43613-7. Zbl 0725.20028. Retrieved 2023-11-18.
^Davis, Michael W. (2007). The Geometry and Topology of Coxeter Groups(PDF). Princeton University Press. ISBN 978-0-691-13138-2. Zbl 1142.20020. Retrieved 2023-11-18.
In mathematics, a Coxetergroup, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic...
In mathematics, a Coxeter element is an element of an irreducible Coxetergroup which is a product of all simple reflections. The product depends on the...
n Coxetergroup has n mirrors and is represented by a Coxeter–Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram...
reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxetergroups, and are important...
infinite facets whose quotient group of their normal abelian subgroups is finite. They include the one-dimensional Coxetergroup I ~ 1 {\displaystyle {\tilde...
mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxetergroup. Coxeter complexes...
Coxetergroups, so the affine symmetric groups are Coxetergroups, with the s i {\displaystyle s_{i}} as their Coxeter generating sets. Each Coxeter group...
reflection group. Reflection groups also include Weyl groups and crystallographic Coxetergroups. While the orthogonal group is generated by reflections...
Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter...
century. Coxeter was born in Kensington, England, to Harold Samuel Coxeter and Lucy (née Gee). His father had taken over the family business of Coxeter & Son...
context—for example, whether one is discussing general Coxetergroups or complex reflection groups—but in all cases the collection of parabolic subgroups...
theory of Coxeter groups, the symmetric group is the Coxetergroup of type An and occurs as the Weyl group of the general linear group. In combinatorics...
{\displaystyle {\tilde {A}}_{3}} Coxetergroup. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams: The rectified cubic...
special kind of Coxeter diagram), the Weyl group (a concrete reflection group), or the abstract Coxetergroup. Although the Weyl group is abstractly isomorphic...
groups in two dimensions. Other finite subgroups include: Permutation matrices (the Coxetergroup An) Signed permutation matrices (the Coxetergroup Bn);...
Bi=M\wr \mathbb {Z} _{2}.\,} The Bimonster is also a quotient of the Coxetergroup corresponding to the Dynkin diagram Y555, a Y-shaped graph with 16 nodes:...
symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure. Its Coxeter symbol is 421...
passing through the same point are the finite Coxetergroups, represented by Coxeter notation. The point groups in three dimensions are heavily used in chemistry...
Groups of this type are identified by a parameter n, the dimension of the hypercube. As a Coxetergroup it is of type Bn = Cn, and as a Weyl group it...
symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxetergroups or Weyl groups, including...
from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices). Its Coxeter symbol is...
four-dimensional crystal classes 1985 H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, Coxeter notation for 4D point groups 2003 John Conway and Smith, On Quaternions...
be visualized as symmetric orthographic projections in Coxeter planes of the E6 Coxetergroup, and other subgroups. Symmetric orthographic projections...
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