Abstract. In this paper we study complex representations of the factorpower ${{\cal FP}^+}(G,M)$ of a finite group $G$ acting on a finite set $M$. This includes the finite monoid ${{\cal FP}^+(S_n)}$, which can be seen as a kind of ``balanced'' generalization of the symmetric group $S_n$ inside the semigroup of all binary relations. We describe all irreducible representations of ${{\cal FP}^+}(G,M)$ and relate them to irreducible representations of certain inverse semigroups. In particular, irreducible representations of ${{\cal FP}^+(S_n)}$ are related to irreducible representations of the maximal factorizable submonoid of the dual symmetric inverse monoid. We also show that in the latter cases irreducible representations lead to an interesting combinatorial problem in the representation theory of $S_n$, which, in particular, is related to Foulkes' conjecture. Finally, we show that all simple ${{\cal FP}^+}(G,M)$-modules are unitarizable and that tensor products of simple ${{\cal FP}^+}(G,M)$-modules are completely reducible.

AMS Subject Classification (1991): 20M30, 20M18, 20C30

Keyword(s): symmetric group, simple module, factorpower, Foulkes' conjecture, tensor product

Received July 10, 2008, and in revised form Spetember 19, 2008. (Registered under 6424/2009.)