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

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 $F\wr {\mathbf{F}\mathbf{I}}_n$ where ${\bf FI}_{n}$isthecategoryofallinjectivefunctionsbetweensubsetsofan$n$-element set.