Combinatorics and representations of complex reflection groups G(r, p, n)

For every r, n, p|r there is a complex reflection group, denoted G(r, p, n), consisting of all monomial n × n matrices such that all the nonzero entries are r ^th roots of the unity and the r/p ^th power of the product of the nonzero entries is 1. By considering these groups as subgroups of the colored permutation groups, ℤ _r ̃ S _n, we use Clifford theory to define on G(r, p, n) combinatorial parameters and descent representations previously defined on Classical Weyl groups. One of these parameters is the major index which also has an important role in the decomposition of descent representations into irreducibles. We present also a Carlitz identity for these complex reflection groups.