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.

C.F. was partially supported by the Simons Foundation Collaboration Grant for Mathematics #523991. E.G. was partially supported by the National Science Foundation (DMS-1800749). D.G. was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and by Capes-PrInt grant number 88881.310538/2018-01 - Brazil. C.F. also likes to thank the sabbatical program at the University of Colorado-Boulder for support. The authors thank Judith Packer for helpful conversations.