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

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

2006-06-16

arxiv.org/abs/math/0606391v3

SIGMA 2 (2006), 085, 12 pages

Mathematics

Classical Analysis and ODEs

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Sym

Scientific paper

10.3842/SIGMA.2006.085

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).

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