Primitive partial permutation representations of the polycyclic monoids and branching function systems

We characterise the maximal proper closed inverse submonoids of the polycyclic inverse monoids, also known as Cuntz inverse semigroups, and so determine all their primitive partial permutation representations. We relate our results to the work of Kawamura on certain kinds of representations of the Cuntz C∗-algebras and to the branching function systems of Bratteli and Jorgensen.