Reflection Groups and Coxeter Groups

Abstract

This chapter is of an auxiliary nature and contains the modicum of the theory of finite reflection groups and Coxeter groups which we need for a systematic development of the theory of Coxeter matroids. A reflection group W is a finite subgroup of the orthogonal group of ℝⁿ generated by some reflections in hyperplanes (mirrors or walls). The mirrors cut ℝⁿ into open polyhedral cones, called chambers. The geometric concepts associated with the resulting chamber system (called the Coxeter complex of W) form the language of the theory of Coxeter matroids. The reader familiar with the theory of reflection groups and Coxeter groups may skip most of the chapter. However, we recommend that this reader look through Sections 5.12 “Residues,” 5.14 “Bruhat order” and 5.15 “Splitting the Bruhat order.”