Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface.

We consider a non-Hermitian matrix orthogonality on a contour in the complex plane. Given a diagonalizable and rational matrix valued weight, we show that the Christoffel--Darboux (CD) kernel, which is built in terms of matrix orthogonal polynomials, is equivalent to a scalar valued reproducing kernel of meromorphic functions in a Riemann surface. If this Riemann surface has genus $0$, then the matrix valued CD kernel is equivalent to a scalar reproducing kernel of polynomials in the plane. We find that, interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. We provide several applications of our result to the theory of tiling models with doubly periodic weightings. In particular, we show that the correlation kernel of any lozenge tiling model with $2 \times 1$ or $2 \times 2$ periodic weightings admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.