Scalar products of symmetric functions and matrix integrals

Details Scalar products of symmetric functions and matrix integrals Scalar products of symmetric functions and matrix integrals

2002-11-28

arxiv.org/abs/nlin/0211051v3

Theor.Math.Phys. 137 (2003) 1676-1690; Teor.Mat.Fiz. 137 (2003) 375-392

Nonlinear Sciences

Exactly Solvable and Integrable Systems

LaTex file. 15 pgs. Based on talks by J. Harnad and A. Yu. Orlov at the workshop: Nonlinear evolution equations and dynamical

Scientific paper

10.1023/B:TAMP.0000007916.13779.

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.

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