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

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 $\underset{m\ge 0}{⨁}({\mathsf{A}}_m,{\mathsf{A}}_n)-$bimodules. A complete description of the category is an open problem.

20 pages, to appear in Mathematical Physics, MDPI