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We study isometric representations of product systems of correspondences overthe semigroup $\mathbb{N}^k$ which are minimal dilations of finite dimensional,fully coisometric representations. We show the existence of a unique minimalcyclic coinvariant subspace for all such representations. The compression ofthe representation to this subspace is shown to be complete unitary invariant.For a certain class of graph algebras the nonself-adjoint \textsc{wot}-closedalgebra generated by these representations is shown to contain the projectiononto the minimal cyclic coinvariant subspace. This class includes freesemigroup algebras. This result extends to a class of higher-rank graphalgebras which includes higher-rank graphs with a single vertex.

Finitely Correlated Representations of Product Systems of $C^*$-Correspondences over $\mathbb{N}^k$