On the relation between orthogonal, symplectic and unitary matrix ensembles
Abstract
For the unitary ensembles of
N×N
Hermitian matrices associated with a weight function
w
there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are
2×2
matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever
w
′
/w
is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of
w
′
/w
. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations.