Weyl Group - Coxeter Group Structure

Coxeter Group Structure

Weyl groups are examples of finite reflection groups, as they are generated by reflections; the abstract groups (not considered as subgroups of a linear group) are accordingly finite Coxeter groups, which allows them to be classified by their Coxeter–Dynkin diagram.

Concretely, being a Coxeter group means that a Weyl group has a special kind of presentation in which each generator x_i is of order two, and the relations other than x_i2 are of the form (x_ix_j)m_ij. The generators are the reflections given by simple roots, and m_ij is 2, 3, 4, or 6 depending on whether roots i and j make an angle of 90, 120, 135, or 150 degrees, i.e., whether in the Dynkin diagram they are unconnected, connected by a simple edge, connected by a double edge, or connected by a triple edge.

Weyl groups have a Bruhat order and length function in terms of this presentation: the length of a Weyl group element is the length of the shortest word representing that element in terms of these standard generators. There is a unique longest element of a Coxeter group, which is opposite to the identity in the Bruhat order.