Weyl group

A finite reflection group defined as $N_G(T)/T$ (or via root reflections) acting on the Cartan and its dual.

Weyl group

Definition (Lie group version)

Let $G$ be a compact connected Lie group and let $T\subset G$ be a maximal torus (see maximal tori ). The Weyl group of $G$ is

W(G,T)=N_G(T)/T,

where $N_G(T)=\{g\in G\mid gTg^{-1}=T\}$ is the normalizer. It is a finite group, and it acts on $\mathfrak t=\mathrm{Lie}(T)$ by the adjoint action and hence on $\mathfrak t^\ast$ by duality.

Definition (root system version)

For a complex semisimple Lie algebra $\mathfrak g$ with Cartan subalgebra $\mathfrak h$ and root system $\Delta\subset \mathfrak h^\ast$ , the Weyl group can be described as the subgroup of $\mathrm{GL}(\mathfrak h^\ast)$ generated by reflections

s_\alpha(\lambda)=\lambda-\lambda(H_\alpha)\,\alpha

for $\alpha\in\Delta$ , where $H_\alpha$ is the coroot. This realizes $W$ as a finite reflection group attached to the root system .

Why it matters

The Weyl group controls much of the combinatorics of semisimple Lie theory:

it permutes the roots and the weight lattice ;
it identifies different choices of positive roots and simple roots (compare positive roots and simple roots );
it is encoded by the Dynkin diagram and appears in classification and character formulas.