Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 085, 12 pages      math.CA/0606391      https://doi.org/10.3842/SIGMA.2006.085
Contribution to the Vadim Kuznetsov Memorial Issue

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

Hjalmar Rosengren
Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden

Received October 11, 2006; Published online December 04, 2006

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).

Key words: Christoffel-Darboux kernel; multivariable orthogonal polynomial; Pfaffian; determinant; correlation function; random Hermitian matrix; orthogonal polynomial ensemble; Sundquist's identities.

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