Natural transformations between induction and restriction on iterated wreath product of symmetric group of order $2$

TL;DR

This paper investigates the structure of iterated wreath products of symmetric groups of order 2, focusing on their centers, cosets, and natural transformations between induction and restriction functors, with applications to representation theory.

Contribution

It provides explicit descriptions of Mackey theory and hom spaces for these wreath product groups, advancing understanding of their representation categories.

Findings

Structural properties of $ extsf{A}_n$ groups are established.

Explicit Mackey theorem for $ extsf{A}_n$ is derived.

Partial description of natural transformations between induction and restriction is given.

Abstract

Let $\mathbb{C}\mathsf{A}_n = \mathbb{C}[S_2\wr S_2 \wr\cdots \wr S_2]$ be the group algebra of $n$-step iterated wreath product. We prove some structural properties of $\mathsf{A}_n$ such as their centers, centralizers, right and double cosets. We apply these results to explicitly write down Mackey theorem for groups $\mathsf{A}_n$ and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of $\displaystyle \bigoplus_{m\geq 0}(\mathsf{A}_m, \mathsf{A}_n)-$bimodules. A complete description of the category is an open problem.