Irreducibility and monicity for representations of k-graph C* -algebras

The representations of a k-graph C∗ -algebra C∗ (Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. In addition, we study the relationship between monic representations and the periodicity of Λ; our analysis yields results which are new even in the case of directed graphs. Finally, we explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C∗ (Λ). Throughout the paper, we work in the setting of row-finite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality.