Scalar products of symmetric functions and matrix integrals

Abstract

We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions and integrals defining matrix model partition functions. Using the fermionic Fock space representation, a proof of the expansion of an associated class of KP and 2-Toda tau functions ${\tau }_{r,n}$ in a series of Schur functions generalizing the hypergeometric series is given and related to the scalar product formulae. It is shown how special cases of such $\tau$-functions may be identified as formal series expansions of partition functions. A closed form exapnsion of $\log {\tau }_{r,n}$ in terms of Schur functions is derived.

arXiv commentLaTex file. 15 pgs. Based on talks by J. Harnad and A. Yu. Orlov at the workshop: Nonlinear evolution equations and dynamical systems 2002, Cadiz (Spain) June 9-16, 2002. To appear in proceedings. (Minor typographical corrections added, abstract expanded)

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