                     Weyl Groups and Hecke Algebras

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 f"ur Mathematik, RWTH
Aachen,    Templergraben    64,    5100    Aachen,    Germany,    e-mail:
geck@tiffy.math.rwth-aachen.de).

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.
