Three different methods are provided for computing with a Coxeter group: the Coxeter presentation, the permutation representation on roots, or a reflection representation.
For most purposes, the presentation will be the most useful of these descriptions. The standard normal form is used for elements (the lexicographically least word of minimal length). Robert Howlett has implemented his highly efficient method for normalising and multiplying elements, based on ideas from [BH93].
If the Coxeter group is finite, it is often better to use the permutation representation. Note that elements are represented as permutations on the set of roots. This is not the minimal degree representation, but is more useful in many cases.
Finally, Coxeter groups can be represented as a reflection group over the reals (in practice over a number field, since the reals are not infinite precision). Although functions are provided for creating reflection groups over an arbitrary field, fewer facilities are available for such groups. In addition, functions are provided to construct all the finite complex reflection groups.
Efficient functions are provided for converting between these three forms of Coxeter group.
This is described in Chapters COXETER GROUPS and REFLECTION GROUPS.