The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category $F \wr {\bf FI}_n$

TL;DR

This paper presents a new, elementary proof of the Littlewood-Richardson rule for wreath products of finite groups with symmetric groups, and applies it to compute the quiver of a related category, enhancing understanding of their representation theory.

Contribution

It provides a novel proof avoiding symmetric functions and derives a branching rule, extending the Littlewood-Richardson rule to wreath products and categories of injective functions.

Findings

Elementary proof of Littlewood-Richardson rule for wreath products

Branching rule for embeddings of wreath products

Computation of the quiver of the category $F \wr \mathbf{FI}_n$

Abstract

We give a new proof for the Littlewood-Richardson rule for the wreath product $F \wr S_{n}$ where $F$ is a finite group. Our proof does not use symmetric functions but more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of $F\wr S_{n}$ to $F\wr S_{n+1}$. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category $F \wr {\bf FI}_{n}$ where ${\bf FI}_{n}$ is the category of all injective functions between subsets of an $n$-element set.