Coxeter Group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of reflections. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934) as abstractions of reflection groups, and finite Coxeter groups were classified in 1935 (Coxeter 1935).

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.

Standard references include (Humphreys 1990) and (Davis 2007).

Read more about Coxeter GroupDefinition, An Example, Connection With Reflection Groups, Affine Coxeter Groups, Hyperbolic Coxeter Groups, Partial Orders, Homology

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Complex Reflection Group - Classification
... Any real reflection group becomes a complex reflection group if we extend the scalars from R to C ... In particular all Coxeter groups or Weyl groups give examples of complex reflection groups ... Any finite complex reflection group is a product of irreducible complex reflection groups, acting on the sum of the corresponding vector spaces ...
Longest Element Of A Coxeter Group - Properties
... A Coxeter group has a longest element if and only if it is finite "only if" is because the size of the group is bounded by the number of words of length less than or equal to the maximum ... The longest element of a Coxeter group is the unique maximal element with respect to the Bruhat order ... If the Coxeter group is a finite Weyl group then the length of w0 is the number of the positive roots ...
Coxeter Group - Homology
... Since a Coxeter group W is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2-group, i.e ... to the direct sum of several copies of the cyclic group Z2 ... This may be restated in terms of the first homology group of W ...
Longest Element Of A Coxeter Group
... In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of ...

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