Abelianization of Matrix Orthogonal Polynomials

Abstract

The main goal of the paper is to connect matrix polynomial biorthogonality on a contour in the plane with a suitable notion of scalar, multi-point Padé approximation on an arbitrary Riemann surface endowed with a rational map to the Riemann sphere. To this end we introduce an appropriate notion of (scalar) multi-point Padé approximation on a Riemann surface and corresponding notion of biorthogonality of sections of the semicanonical bundle (half-differentials). Several examples are offered in illustration of the new notions.

Matrix Orthogonal Polynomials, non-abelian Toda lattice and Bäcklund transformation

“…It was found that the Hermite-type matrix orthogonal polynomials satisfy a second-order differential equation, which is to admit a double integral representation of the matrix-valued Christoffel-Darboux kernels. Some other examples could be found at [8,34].…”

“…(1.1) This is a non-commutative generalization of standard (bi-)orthogonal polynomials and related matrix-valued moment problem could date back to Krein [35]. In recent years, there have been a few recent publications concerning applications of matrix-valued orthogonal polynomials into different subjects such as approximation problems on Riemann sphere [8], quasi-birth-and-death processes [32] and random tiling problems [23]. Among those, the relation between matrix orthogonal polynomials and integrable systems is always a key topic [5,11,12,34,40].…”

“…b exp(xt) • 1 0<x<∞ dx,with the same constant in off-diagonal part and 1 0<x<∞ is the indicator function which is taken as 1 when 0 < x < ∞ and 0 otherwise. More examples could be found at[8, Sec. 4.1.1] and [11, Example 4.1].…”

Biased $2 \times 2$ periodic Aztec diamond and an elliptic curve

“…For the biased model it is interesting to see what the flow on the elliptic curve implies for the matrixvalued orthogonal polynomials, and if explicit expressions can be given in general and/or for the periodic case. Furthermore, it is interesting to compare our results with [6], in which matrix orthogonal polynomials were studied using an abelization based on the spectral curve for the orthogonality weight.…”