Coxeter and Reflection Groups

We provide three different methods for computing with a Coxeter group: the presentation, a permutation representation, or a reflection representation.

For most purposes, the presentation will be most useful. We use the standard normal form for elements (the lexicographically least word of minimal length). Robert Howlett has implemented his highly efficient method for normalising and multiplying elements.

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 (the minimal degree representation can be computed using StandardActionGroup).

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). We also provide functions for creating reflection groups over an arbitrary field, although fewer facilities are available for such groups. In particular, we have functions to construct all the finite complex reflection groups.

We also provide efficient functions to convert between these three forms of Coxeter group. In particular, every Coxeter group is assumed to have a fixed underlying reflection representation.