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

The representations of a $k$-graph $C^*$-algebra $C^*(Lambda)$ which arise from $Lambda$-semibranching function systems are closely linked to the dynamics of the $k$-graph $Lambda$. 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. We also explore the relationship between irreducible $Lambda$-semibranching representations and purely atomic representations of $C^*(Lambda)$. Throughout the paper, we work in the setting of row-finite source-free $k$-graphs; this paper constitutes the first analysis of $Lambda$-semibranching representations at this level of generality.