Introduction

This chapter describes Magma functions for computing with finite Coxeter groups. It was implemented by Scott Murray and Don Taylor, modelled partly on the Chevie package in GAP [GHLetalchar+96].

These groups can be represented in Magma in three ways:

• As a permutation group acting on the roots of the root datum (Chapter ROOT DATA FOR LIE THEORY). This is the main subject of this chapter. Permutation groups of this kind are in the category GrpCox, which is a subcategory of GrpPerm. If H is a Coxeter subgroup of the Coxeter group W, then the elements of H permute the roots of W.

• As a finitely presented group with the standard Coxeter group presentation (Section Finitely presented Coxeter groups). We will call this an FP Coxeter group, to distinguish it from the Coxeter groups of item (1).

• As the matrix group acting on the (co)root space (Section Actions on roots and coroots). We call this a (co)reflection group.