Hopf Modules in the Representation Theory of Wreath Products

TL;DR

This paper explores the structure of Hopf modules in the representation theory of wreath product groups, providing explicit formulas for algebra morphisms and analyzing their interactions through induction and restriction.

Contribution

It introduces a class of Hopf modules over PSH-algebras with a compatibility condition and derives explicit formulas for morphisms between these algebras.

Findings

Characterization of Hopf modules over PSH-algebras

Explicit formulas for induction and restriction morphisms

Analysis of restricted wreath product subgroups

Abstract

For a finite group G one may consider the associated tower S_n[G] of wreath product groups. Zelevinsky associates to such a tower a positive self-adjoint Hopf algebra (PSH-algebra) R(G) as the infinite direct sum of the Grothendieck groups of the categories of complex representations of these groups. In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) for subgroups H of G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k^{th}-power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms between R(H) and R(G) arising from induction and restriction. In an appendix, we consider the more general family of restricted wreath products, which are subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n).