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Scalar products of symmetric functions and matrix integrals

J. Harnad·A. Yu. Orlov
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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 $τ_{r,n}$ 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 $τ$-functions may be identified as formal series expansions of partition functions. A closed form exapnsion of $\log τ_{r,n}$ in terms of Schur functions is derived.

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J. Harnad, A. Yu. Orlov. 2002-12-10. Scalar products of symmetric functions and matrix integrals. https://doi.org/10.1023/b%3Atamp.0000007916.13779.17

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