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Concept
Coxeter group
Summary
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; 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 in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 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
and .
Formally, a Coxeter group can be defined as a group with the presentation
where and for .
The condition means no relation of the form should be imposed.
The pair where is a Coxeter group with generators is called a Coxeter system. Note that in general is not uniquely determined by . For example, the Coxeter groups of type and are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation).
A number of conclusions can be drawn immediately from the above definition.
The relation means that for all ; as such the generators are involutions.
If , then the generators and commute. This follows by observing that
together with
implies that
Alternatively, since the generators are involutions, , so , and thus is equal to the commutator.
In order to avoid redundancy among the relations, it is necessary to assume that . This follows by observing that
together with
implies that
Alternatively, and are conjugate elements, as .
The Coxeter matrix is the symmetric matrix with entries .
Official source
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