Recall that the reflections ${s_{\alpha} \in Aut(E)}$ for ${\alpha \in \Phi}$ preserve ${\Phi}$. They generate a group ${W}$ called the Weyl group. Moreover, since ${\Phi}$ spans ${E}$, the map ${W \rightarrow S_{\Phi}}$, the symmetric group on ${\Phi}$, is injective. So ${W}$ is a finite group of orthogonal isomorphisms of ${E}$, i.e. leaving invariant the bilinear form ${(\cdot, \cdot)}$.

Everything here actually makes sense for root systems in general, but we are restricting ourselves to the case of a root system associated to semisimple Lie algebra and a Cartan subalgebra.  The only difference is that one has to prove the result on maximal strings (which was proved in the case of Lie algebras here), though it can be done for root systems in general.

Now choose a base ${\Delta}$ for ${\Phi}$ and a corresponding partition ${\Phi = \Phi^+ \cup \Phi^-}$; the ${s_{\delta}}$ for ${\delta \in \Delta}$ are called simple reflections.

The goal we are aiming for is the following theorem, which gives a large amount of information about the Weyl group.

Theorem 1 ${W}$ acts simply transitively on Weyl chambers and on bases. Any root can be moved by an element of the Weyl group into a given base. ${W}$ is generated by the simple reflections. (more…)