Induction/Restriction Bialgebras for Restricted Wreath Products

Abstract

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].