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 (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 (Coxeter 1934) as abstractions of reflection groups, and finite Coxeter groups were classified in 1935 (Coxeter 1935). GO 6 − ⁡ ( 2 ) ≅ PSU 4 ⁡ ( 2 ) : 2 ≅ SO 5 ⁡ ( 3 ) ≅ Sp 4 ⁡ ( 3 ) {displaystyle {egin{aligned}operatorname {GO} _{6}^{-}(2)&cong operatorname {PSU} _{4}(2)colon 2\&cong operatorname {SO} _{5}(3)\&cong operatorname {Sp} _{4}(3)end{aligned}}} 221, 122 D 2 n {displaystyle D_{2n}} TriangleSquareTesseractDemitesseract 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 (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 1992) and (Davis 2007). Formally, a Coxeter group can be defined as a group with the presentation where m i i = 1 {displaystyle m_{ii}=1} and m i j ≥ 2 {displaystyle m_{ij}geq 2} for i ≠ j {displaystyle i eq j} .The condition m i j = ∞ {displaystyle m_{ij}=infty } means no relation of the form ( r i r j ) m {displaystyle (r_{i}r_{j})^{m}} should be imposed. The pair ( W , S ) {displaystyle (W,S)} where W {displaystyle W} is a Coxeter group with generators S = { r 1 , … , r n } {displaystyle S={r_{1},dots ,r_{n}}} is called a Coxeter system. Note that in general S {displaystyle S} is not uniquely determined by W {displaystyle W} . For example, the Coxeter groups of type B 3 {displaystyle B_{3}} and A 1 × A 3 {displaystyle A_{1} imes A_{3}} are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation).