  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).
