Weyl group

The Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. For example, the root system of A₂ consists of the vertices of a regular hexagon centered at the origin. The full group of symmetries of this root system is therefore the dihedral group of order 12. The Weyl group is generated by reflections through the lines bisecting pairs of opposite sides of the hexagon; it is the dihedral group of order 6.

The Weyl group of a semi-simple Lie group, a semi-simple Lie algebra, a semi-simple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.

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 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 chamber, the set &Phi⁺, 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}.

Weyl groups are examples of Coxeter groups. This means that they have 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. The length of a Weyl group element is the length of the shortest word representating that element in terms of these standard generators.

If G is a semisimple linear algebraic group over an algebraically closed field (more generally a split group), and T is a maximal torus, the normalizer N of T contains T as a subgroup of finite index, and the Weyl group W of G is isomorphic to N/G. If B is a Borel subgroup of G, i.e., a maximal connected solvable subgroup and T is chosen to lie in B, then we obtain the Bruhat decomposition

which gives rise to the decomposition of the flag variety G/B into Schubert cells (see Grassmannian).