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

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