Finite Coxeter Groups and the Weak Order

Abstract

In this chapter, we develop the basic theory of finite Coxeter groups, drawing on results already proved for posets of regions. There are two main points to this chapter: First, to show how the geometry and lattice theory of hyperplane arrangements underlies the theory of finite Coxeter groups, and second, to point out the weak orders on finite Coxeter groups as an important class of lattice-theoretic examples. A broader class of examples is obtained as lattice quotients of weak orders. Several examples of such quotients are given in Sections 10-6 and 10-7.