Generators for the representation rings of certain wreath products

TL;DR

This paper generalizes the generation of representation rings from symmetric groups to wreath products with finite groups or Hopf algebras within Deligne categories, confirming a conjecture for Coxeter groups of type B.

Contribution

It extends Marin's result to wreath products and verifies a conjecture for Coxeter groups of type B in the Deligne category framework.

Findings

Representation rings of wreath products are generated by specific elements.

The generalization applies to finite groups and Hopf algebras.

Confirmed Marin's conjecture for Coxeter groups of type B.

Abstract

Working in the setting of Deligne categories, we generalize a result of Marin that hooks generate the representation ring of symmetric groups to wreath products of symmetric groups with a fixed finite group or Hopf algebra. In particular, when we take the finite group to be cyclic order 2 we recover a conjecture of Marin about Coxeter groups in type B.