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3.3 Finite Coxeter groups

An important class of Coxeter groups are the finite ones, like I2(3) above. One can show that a Coxeter group is finite if and only if the scalar product defined by Equation (3.28View Equation) on V is Euclidean [107Jump To The Next Citation Point]. Finite Coxeter groups coincide with finite reflection groups in Euclidean space (through hyperplanes that all contain the origin) and are discrete subgroups of O (n). The classification of finite Coxeter groups is known and is given in Table 1 for completeness. For finite Coxeter groups, one has the important result that the Tits cone coincides with the entire space V [107Jump To The Next Citation Point].


Table 1: Finite Coxeter groups.

Name

Coxeter graph

An

PIC

   

Bn ≡ Cn

PIC

   

Dn

PIC

   

I2(m )

PIC

   

F4

PIC

   

E6

PIC

   

E7

PIC

   

E8

PIC

   

H3

PIC

   

H4

PIC



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