Dear Prof. Neil,
in GAP-3.2 there is a package containing programs for working with
Weyl groups and associated Hecke algebras. The latter are certain deformations
of the group ring of the first. There are programs to do computations
in this algebra, e.g. multiplying two basis elements and expressing the
result as a linear combination of the basis vectors. (I am not certain
whether or not GAP has already some built-in functions for comptations
in group rings). Maybe looking at the code or the documented examples
in the chapter 'Weyl group and Hecke algebras' of the manual might help you?

Sincerely yours, Meinolf Geck

(Lehrstuhl D f"ur Mathematik, RWTH Aachen)