Weyl groups

Abstract

Weyl groups are integral forms for finite Euclidean reflection groups. Their data consist of the reflection group structure, plus a lattice in Euclidean space that is stable under the group action. In this chapter, we shall show that a reflection group has such an equivariant lattice if and only if it has a crystallographic root system. Moreover, a definite relation will also be established between the crystallographic root system and the possible equivariant lattices. This, then, reduces the classification of Weyl groups to that of crystallographic root systems, and that classification will be carried out in the next chapter.