Author: John Stembridge (jrs@math.lsa.umich.edu), Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003
From the Author:
Coxeter and Weyl are two packages of~37 Maple programs that provide assistance in the study of root systems and finite Coxeter groups. The coxeter package contains 30 basic procedures for working with the roots themselves, for computing reduced expressions for group elements, generating permutation representations and characters, computing various combinatorial invariants, such as generating functions for length, and retrieving information such as what appears in the appendices of N. Bourbaki's Groupes et Algèbres de Lie, Chp. IV--VI. The weyl package is a supplement to the coxeter package that contains 7 procedures for manipulating weight vectors and computing weight multiplicities for irreducible representations of semisimple Lie algebras. It is capable of reproducing essentially the entire contents of the book Tables of dominant weight multiplicities for representations of simple Lie algebrasby Bremner, Moody and Patera.
I'm not about to claim that these packages will answer every question one might want to know about a root system or Coxeter group. However, even if it doesn't do exactly what you want, there is a good chance you can build the procedures you need out of the raw materials provided here. Also, I do not intend to claim that the algorithms used here represent spectacular advances over what is already known. And don't send me complaints if you think it is too slow. If speed is a major consideration, then you probably shouldn't be using Maple---a better alternative would be to use the LiE package from CWI.