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Abstract When the reduced twisted $C^*$-algebra $$C^*_r({\mathcal{G}}, c)$$ of a non-principal groupoid $${\mathcal{G}}$$ admits a Cartan subalgebra, Renault’s work on Cartan subalgebras implies the existence of another groupoid description of $$C^*_r({\mathcal{G}}, c)$$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $${\mathcal{S}}$$ of $${\mathcal{G}}$$. In this paper, we study the relationship between the original groupoids $${\mathcal{S}}, {\mathcal{G}}$$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $${\mathfrak{B}}$$ of the Cartan subalgebra $$C^*_r({\mathcal{S}}, c)$$. We then show that the quotient groupoid $${\mathcal{G}}/{\mathcal{S}}$$ acts on $${\mathfrak{B}}$$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map $${\mathcal{G}}\to{\mathcal{G}}/{\mathcal{S}}$$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $$2$$-cocycle on $${\mathcal{G}}/{\mathcal{S}} \ltimes{\mathfrak{B}}$$. 

In this paper, we define the notion of monic representation for the $$C^{\ast }$$ -algebras of finite higher-rank graphs with no sources, and we undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative $$C^{\ast }$$ -algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the $$\unicode[STIX]{x1D6EC}$$ -semibranching representations previously studied by Farsi, Gillaspy, Kang and Packer (Separable representations, KMS states, and wavelets for higher-rank graphs. J. Math. Anal. Appl.   434  (2015), 241–270) and also provide a universal representation model for non-negative monic representations.