Coxeter Group - Definition

Definition

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 (W,S) where W is a Coxeter group with generators S={r1,...,rn} is called Coxeter system. Note that in general S is not uniquely determined by W. For example, the Coxeter groups of type BC3 and A1xA3 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 mi i = 1 means that (riri )1 = (ri )2 = 1 for all i ; the generators are involutions.
  • If mi j = 2, then the generators ri and rj commute. This follows by observing that
xx = yy = 1,
together with
xyxy = 1
implies that
xy = x(xyxy)y = (xx)yx(yy) = yx.
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 mi j = mj i. This follows by observing that
yy = 1,
together with
(xy)m = 1
implies that
(yx)m = (yx)myy = y(xy)my = yy = 1.
Alternatively, and are conjugate elements, as .

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