Jean,
Ok, I will implement the method at the level of
Coxeter groups and use reflections.
For Weyl groups I will put in a method that
converts reflections to positive (co)roots.
Concerning the numbering of reflections:
For infinite Coxeter groups (including affine Weyl groups,
which I use a lot
Just a remark.
The list of inversions, in my view, should preferably be a list of
reflections (which does not need the existence of roots and makes sense for
abstract Coxeter groups).
That is, for w=s_1...s_n, the list of right inversions is the list
s_n, s_n s_{n-1} s_n, ... , s_
