Preliminary support for complex reflection groups and cyclotomic Hecke
algebras has been added to the CHEVIE package. A complex reflection
group is a group W acting on a vector space V, and generated by
pseudo-reflections in V. The field of definition of W is defined to
be the field of definition of V. It turns out that, as for rational
reflection groups (Weyl groups), all representations of a complex
reflection group W are defined over the field of definition of W (cf.
Ben76 and D.~Bessis thesis). Similarly to Coxeter groups, complex
reflection groups are represented by the permutation representation on a
set of roots in V invariant by W and such that all reflections in
W are reflections with respect to some root (see
ComplexReflectionGroup). However there is no general theory on how to
construct a nice set of roots for an arbitrary reflection group; the
roots given in GAP where obtained case-by-case in an ad hoc way.
Irreducible complex reflection groups have been classified by Shephard
and Todd. They contain one infinite family depending on 3 parameters,
and 34 ``exceptional'' groups (which have been given by Shephard and
Todd a number which actually varies from 4 to 37, and covers also the
exceptional Coxeter groups, e.g., CoxeterGroup("E",8) is the group of
Shephard-Todd number 37).
The cyclotomic Hecke algebra (see Hecke for complex reflection groups) corresponding to a complex reflection group is defined in a similar way as the Iwahori--Hecke algebra; for details see BM93. G.~Malle has computed character tables for some of these algebras, including all 2-dimensional groups, see BM93 and Mal96; CHEVIE contains those of type G(e,1,1), G_4, G_5, G_6, G_8, G_9, G_{12} and G_{25} in the Shephard-Todd classification.
GAP 3.4.4