Classification of Coxeter systems and reflection groups

Abstract

In this chapter, we obtain a classification of finite Coxeter systems and finite Euclidean reflection groups. We have already shown in Chapter 6 that every reflection group has a canonical associated Coxeter system. So the classifications are related. In Chapter 7 we introduced the bilinear form of a Coxeter system. Most of this chapter is occupied with determining necessary conditions for the bilinear form B: V × V → ℝ of a finite Coxeter system (W, S) to be positive definite. We then have strong restrictions on the possibilities for finite Coxeter systems. The classification is finished by proving that each of these remaining possibilities can be realized by the case of a finite reflection group.