Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices

Abstract

We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as sums of squares).

Keywords

• Christoffel-Darboux kernel
• multivariable orthogonal polynomial
• Pfaffian
• determinant
• correlation function
• random Hermitian matrix
• orthogonal polynomial ensemble
• Sundquist's identities