Weyl Group - Weyl Chambers

Removing the hyperplanes defined by the roots of Φ cuts up Euclidean space into a finite number of open regions, called Weyl chambers. These are permuted by the action of the Weyl group, and it is a theorem that this action is simply transitive. In particular, the number of Weyl chambers equals the order of the Weyl group. Any non-zero vector v divides the Euclidean space into two half-spaces bounding the hyperplane v∧ orthogonal to v, namely v+ and v−. If v belongs to some Weyl chamber, no root lies in v∧, so every root lies in v+ or v−, and if α lies in one then −α lies in the other. Thus Φ+ := Φ∩v+ consists of exactly half of the roots of Φ. Of course, Φ+ depends on v, but it does not change if v stays in the same Weyl chamber. The base of the root system with respect to the choice Φ+ is the set of simple roots in Φ+, i.e., roots which cannot be written as a sum of two roots in Φ+. Thus, the Weyl chambers, the set Φ+, and the base determine one another, and the Weyl group acts simply transitively in each case. The following illustration shows the six Weyl chambers of the root system A₂, a choice of v, the hyperplane v∧ (indicated by a dotted line), and positive roots α, β, and γ. The base in this case is {α,γ}.