Enumerating Representations in Finite Wreath Products

✍ Scribed by Thomas Müller

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 275 KB
• Volume: 153
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1998.1885

✦ Synopsis

Let 1 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 n , the Weyl group of type D n ). We compute the exponential generating function for the number |Hom(1, R n )| of 1-representations in R n and for some refinements of this sequence under very mild finiteness assumptions on 1 (always met for instance if 1 is finitely generated). This generalizes in a uniform way the connection between the problem of counting finite index subgroups in a group 1 and the enumeration of 1-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 n on the other. We also study the asymptotics of the function |Hom(G, H " S n )| for arbitrary finite groups G and H.