ENUMERATING REPRESENTATIONS IN FINITE WREATH PRODUCTS

THOMAS MÜLLER

Abstract. Let Γ be a group (finite or infinite), H a finite group, and let R_{n} denote the sequence H ≀ S_{n} of symmetric wreath products as well as certain variants of it (including in particular H ≀ A_{n} and W (D_{n}), the Weyl group of type D_{n}). We compute the exponential generating function for the number |Hom(Γ, R_{n})| of Γrepresentations in R_{n} and for some refinements of this sequence under very mild finiteness assumptions on Γ (always met for instance if Γ is finitely generated). This generalizes in a uniform way the connection between the problem of counting finite index subgroups in a group Γ and the enumeration of Γ-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 ≀ S_{n}, H ≀ A_{n}, and W (D_{n}) on the other. We also study the asymptotics of the function |Hom(G, H ≀ S_{n})| for arbitrary finite groups G and H.