Induction/Restriction Bialgebras for Restricted Wreath Products

TL;DR

This paper generalizes the algebraic structures associated with wreath product groups, introducing restricted wreath products and analyzing their representation theory, which extends known decompositions from full to restricted cases.

Contribution

It introduces and studies a new algebra/coalgebra structure for restricted wreath products, extending the Hopf algebra framework to these groups and their representation theory.

Findings

The algebra contains all irreducible representations as constituents.

It generalizes tensor product decompositions from full to restricted wreath products.

The structure is not a Hopf algebra but still contains rich algebraic information.

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].