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,...,r_n} 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 m_{i i} = 1 means that (r_ir_i )1 = (r_i )2 = 1 for all i ; the generators are involutions.
• If m_{i j} = 2, then the generators r_i and r_j 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 m_{i j} = m_{j 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 .