Enumerating representations in finite wreath products
Thomas Müller
Let $\Gamma$ be a group (finite or infinite), $H$ a finite group, and let $R_n$ denote the sequence $H \wr S_n$ of symmetric wreath products as well as certain variants of it (including in particular $H \wr A_n$ and $W(D_n)$, the Weyl group of type $D_n$). We compute the exponential generating function for the number $|\text{Hom}(\Gamma, R_n)|$ of $\Gamma$-representations in $R_n$ and for some refinements of this sequence under very mild finiteness assumptions on $\Gamma$ (always met for instance if $\Gamma$ is finitely generated). This generalizes in a uniform way the connection between the problem of counting finite index subgroups in a group $\Gamma$ and the enumeration of $\Gamma$-actions on finite sets on the one hand, and the recent results of Chigira concerning solutions of the equation $x^m = 1$ in the groups $H \wr S_n$, $H \wr A_n$, and $W(D_n)$ on the other. We also study the asymptotics of the function $|\text{Hom}(G, H \wr S_n)|$ for arbitrary finite groups $G$ and $H$.