Root Systems

In this chapter we introduce objects that are in a sense “dual” to mirror systems: given a mirror system, we take, for each mirror, a pair of normal vectors. If lengths of these normal vectors are chosen in a coordinated way, the resulting system of vectors is preserved by all reflections; in that case we say that we have a root system.

Root systems provide a more traditional approach to finite reflection groups—and have exceptionally important applications on their own. In our approach, we freely use both mirrors and roots with the aim of maximizing the intuitive geometric aspect of the theory.