There is a collection of programs for dealing with finite Weyl groups, associated (Iwahori-) Hecke algebras, and their representations. These programs were written by Meinolf Geck (Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 5100 Aachen, Germany, e-mail: geck@tiffy.math.rwth-aachen.de) in the GAP language.

On a low level, these programs provide the basic data about a fixed finite Weyl group W: having specified a Cartan matrix, they compute the root system, the reflection representation, and the permutation representation on the root vectors. Elements of W can be given either as permutations or as reduced words in the standard generators (i.e., simply as a list of integers labelling the generators), and there are functions to convert these expressions into each other. In particular, all functions of GAP for working with permutation groups can be applied.

On a higher level, it is possible to compute distinguished coset representatives and representatives of the conjugacy classes of $W$ of minimal length. Furthermore, there are functions for calculating Kazhdan-Lusztig polynomials, left cells, and the corresponding representations of the associated Hecke algebra H (for groups of rank <= 5, say). Also, computations inside H are possible, such as multiplying two arbitrary elements of H and expressing the result as a linear combination in the basis elements T_w, w in W.

See also the relevant part of the GAP Manual.

Special Purpose Systems