Weyl group

Remarks

The group $W(\Phi)$ is finite: each element permutes the finite set $\Phi$, and an element that fixes every root is the identity, since $\Phi$ spans $E$. Example: for $A_{n-1}$, the roots $\varepsilon_i - \varepsilon_j$ in the hyperplane $x_1 + \cdots + x_n = 0$ of $\R^n$ give reflections that swap two coordinates, so $W(A_{n-1})$ is the symmetric group $S_n$ acting by permuting coordinates. Non-example: the rotation of the hexagon $A_2$ by $60^\circ$ maps $\Phi$ onto $\Phi$, but it is not in $W(A_2) \cong S_3$. It is $-\id$ composed with a rotation by $240^\circ$, and $-\id$ is not a permutation of the coordinates of the plane $x_1 + x_2 + x_3 = 0$. For a base $\Delta$ of $\Phi$, the simple reflections $s_\alpha$ with $\alpha \in \Delta$ already generate $W(\Phi)$, and they make it a Coxeter group. For a compact connected Lie group, the Weyl group of a compact Lie group is the Weyl group of its root system.