Simple modules over factorpowers

Abstract

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 $\FP$, which can be seen as a kind of a ``balanced'' generalization of the symmetric group $S_n$ 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 $\FP$ 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 $\fp(G,M)$-modules are unitarizable and that tensor products of simple $\fp(G,M)$-modules are completely reducible.