Simple Modules over Factorpowers

In this paper we study complex representations of the factorpower FP(G,M) of a finite group G acting on a finite set M . This includes the finite monoid FP+(Sn), which can be seen as a kind of a “balanced” generalization of the symmetric group Sn inside the semigroup of all binary relations. We describe all irreducible representations of FP(G,M) and relate them to irreducible representations of certain inverse semigroups. In particular, irreducible representations of FP+(Sn) 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 Sn, which, in particular, is related to Foulkes’ conjecture. Finally, we show that all simple FP(G,M)-modules are unitarizable and that tensor products of simple FP(G,M)-modules are completely reducible.