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

TL;DR

This paper explores the relationship between matrix orthogonality in the complex plane and scalar orthogonality on Riemann surfaces, revealing conditions under which matrix kernels simplify to scalar kernels, with applications to tiling models.

Contribution

It establishes an equivalence between matrix orthogonal polynomial kernels and scalar reproducing kernels on Riemann surfaces, especially for genus zero surfaces, simplifying complex correlation kernel formulas.

Findings

Matrix orthogonal kernels are equivalent to scalar kernels on Riemann surfaces.

For genus zero surfaces, matrix kernels reduce to scalar polynomial kernels in the plane.

Application to lozenge tiling models yields simplified double contour integral formulas.

Abstract

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. Interestingly, this scalar reproducing kernel is not necessarily a scalar CD kernel. As an application of our result, we show that the correlation kernel of certain doubly periodic lozenge tiling models admits a double contour integral representation involving only a scalar CD kernel. This simplifies a formula of Duits and Kuijlaars.