In this talk I will explain how the representation theory of the Hecke algebras of the complex reflection groups of type G(r,p,n) is completely determined by the representation theory of Ariki-Koike algebras, or the Hecke algebras of type G(r,1,n), when the parameters for the Hecke algebras are (ε,q)-separated. More precisely, when the parameters are (ε,q)-separated there is an explicit algorithm for computing the decomposition numbers of the algebras of type G(r,p,n) from the decomposition matrices of the algebras of type G(r,1,n).
The proof of this result relies on two Morita equivalences: the first reduces the calculation of all decomposition numbers to the case of the l-splittable decomposition numbers and the second Morita equivalence allows us to compute these decomposition numbers using an analogue of the cyclotomic Schur algebras for the Hecke algebras of type G(r,p,n). We then explicitly compute all of the l-splittable decomposition numbers using detailed calculations with some natural trace functions.
In proving these results, we develop a Specht module theory for these algebras, explicitly construct their simple modules and introduce and study analogues of the cyclotomic Schur algebras of type G(r,p,n) when the parameters are (ε,q)-separated.
This is joint work with Jun Hu.