Primitive representations of the polycyclic monoids and branching function systems

We generalise the group theoretic notion of a primitive permutation representation to inverse monoids and so obtain a notion of a primitive representation by partial permutations. Such representations are shown to be determined by what we call essentially maximal proper closed inverse submonoids. Such submonoids in the case of the polycyclic inverse monoids (also known as Cuntz inverse semigroups) are characterised and all primitive representations of the polycyclic monoids determined. 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. 2000 Mathematics Subject Classification: 20M18, 20M30, 46L05, 20B15.