8 - Complements

Summary

In this final chapter, we survey (without proofs) some related topics which may stimulate the reader to do further reading in the extensive literature of Coxeter groups. These deal with such matters as the internal structure of the groups, their representations, and the Bruhat ordering. The selection of topics and the order of presentation are somewhat random, with no claims of balance or completeness intended. Unless otherwise stated, (W, S) denotes an arbitrary Coxeter system.

The Word Problem

As noted at the end of 5.13, the concrete action of W on a fundamental domain in the dual of the vector space V affording its geometric representation could in principle be used to test which words in the generating set S are equal to 1. But this is extremely cumbersome in practice. Even if programmed for a computer, serious round-off problems can be anticipated, since it is essential to decide whether certain calculated coefficients are strictly positive.

A more attractive method was devised by Tits [5]. It allows one to transform an arbitrary product of generators from S into a reduced expression by making only the most obvious types of modifications coming from the denning relations. Here is a brief description, in our own notation. (For a nice reformulation of Tits' arguments, see pages 49–52 of Brown [1].)

Let F be a free group on a set ∑ in bijection with S (with σ corresponding to s), and let π : F → W be the resulting epimorphism.