# Schur function expansions of matrix integrals

## A. Yu. Orlov

**Abstract: **
We consider the Schur function expansion of various matrix
integrals.

(Based partly on joint work with J.Harnad, T. Shiota and
further developments.)

(1) We consider two-matrix models with an
interaction term that generalizes the standard one, analyzed by
Itsykson and Zuber.

(2) We also consider Schur function expansion for the
partition function of the model of normal matrices. We show that
this expansion coincides with the Takasaki expansion for
tau functions of the Toda lattice hierarchy. We show that the
partition function of the model of normal matrices is, at the
same time, a partition function of certain discrete models, which
can be solved by the method of orthogonal polynomials. We obtain
discrete versions of various known matrix models: models of
non-negative matrices, unitary matrices, normal matrices.

(The latter is based on a joint paper with T.Shiota)