Scalar Products of Symmetric Functions and Matrix Integrals |
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| Authors: | Harnad J Orlov A Yu |
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| Institution: | (1) Centre de Recherches Mathématiques, Université de Montréal, C. P. 6128, succ. centre ville, Montreal, Quebec, Canada, H3C 3J7;(2) Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke W., Montreal, Quebec, Canada, H4B 1R6;(3) Nonlinear Wave Processes Laboratory, Oceanology Institute, Nakhimovskii Prospect 36, Moscow, 117851, Russia |
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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, we prove an expansion of an associated class of KP and 2-Toda tau functions 
r,n
in a series of Schur functions generalizing the hypergeometric series and relate it to the scalar product formulas. We show how special cases of such tau functions can be identified as formal series for partition functions. A closed form expansion of log
r,n
in terms of Schur functions is derived. |
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| Keywords: | symmetric functions hypergeometric functions statistical sums tau functions matrix models Toda lattices |
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