  To a finite group G one can associate a tower of wreath products S_n[G]. It
is well known that the graded direct sum of the Grothendieck groups of the
categories of finite dimensional complex representations of these groups can be
given the structure of a graded Hopf algebra, and in fact a positive
self-adjoint Hopf algebra in the sense of Zelevinsky [1], using the induction
product and restriction coproduct. This paper introduces and explores an
analogously defined algebra/coalgebra structure associated to a more general
class of towers of groups, obtained as a certain family of subgroups of wreath
products in the case G is abelian. We call these groups restricted wreath
products, and they include the infinite family of complex reflection groups
G(m, p, n). It is known that in the case of full wreath products the associated
Hopf algebra decomposes as a tensor power of the Hopf algebra of integral
symmetric functions. In the case of restricted wreath products, the associated
algebra/coalgebra is no longer a Hopf algebra, but here we see that it contains
an algebra containing every irreducible representation as a constituent and
which is isomorphic to a tensor power of such an algebra/coalgebra associated
to a smaller restricted wreath product, generalizing the tensor product
decomposition for the full wreath products. We closely follow the approach of
[1].
