Abstract
We consider a certain scalar product of symmetric functions which is parameterized by a function r and an integer n. On the one hand we have a fermionic representation of this scalar product. On the other hand we get a representation of this product with the help of multi-integrals. This gives links between a theory of symmetric functions, soliton theory and models of random matrices (such as a model of normal matrices).
Similar content being viewed by others
References
Adler, M., Shiota, T. and van Moerbeke, P.: Phys. Lett. A 194(1–2) (1994), 33–34.
Adler, M. and van Moerbeke, P.: Integrals over Grassmannians and random permutations, math.CO/0110281.
Chau, L.-L. and Zaboronsky, O.: On the structure of correlation functions in the normal matrix models, Comm. Math. Phys. 196 (1998), 203–247; hep-th/9711091.
Date, E., Jimbo, M., Kashiwara, M. and Miwa, T.: Transformation groups for soliton equations, In: M. Jimbo and T. Miwa (eds), Nonlinear Integrable Systems – Classical Theory and Quantum Theory, World Scientific, 1983, pp. 39–120.
Dickey, L. A.: Soliton Equations and Hamiltonian System, World Scientific, Singapore, 1991.
Dickey, L. A.: Modern Phys. Lett. A 8 (1993), 1357–1377.
Dryuma, V.: JETF Lett. 19 (1974), 219.
Gross, K. I. and Richards, D. S.: Special functions of matrix arguments. I: Algebraic induction, zonal polynomials, and hypergeometric functions, Trans. Amer. Math. Soc. 301 (1987), 781–811.
Harish Chandra: Amer. J. Math. 80 (1958), 241.
Harnad, J. and Orlov, A. Yu.: Schur function expansions of matrix integrals, CRM Preprint, Montreal, 2001.
Harnad, J. and Orlov, A. Yu.: Scalar products of symmetric functions and matrix integrals, nlin.SI/0211051, reported at the NEEDS-2002 meeting in Cadiz, to be published in Teor. Mat. Phys. 137(2) (November 2003).
Itzykson, C. and Zuber, J. B.: J. Math. Phys. 21 (1980), 411.
Kadomtsev, V. V. and Petviashvili, V. I.: Soviet Phys. Dokl. 15 (1970), 539.
Konopelchenko, B. and Alonso, L. M.: The KP hierarchy in Miwa coordinates, solvint/9905005.
Macdonald, I. G.: Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995.
Mehta, M. L.: Random Matrices, Academic Press, Inc., 1991.
Mikhailov, A. V.: On the integrability of two-dimensional generalization of the Toda lattice, Letters in Journal of Experimental and Theoretical Physics 30 (1979), 443–448.
Milne, S. C.: Summation theorems for basic hypergeometric series of Schur function argument, In: A. A. Gonchar and E. B. Saff (eds), Progress in Approximation Theory, Springer-Verlag, New York, 1992, pp. 51–77.
Mineev-Weinstein, M., Wiegmann, P. and Zabrodin, A.: Integrable structure of interface dynamics, LAUR-99-0703.
Miwa, T.: On Hirota’s difference equations, Proc. Japan Acad. Ser. A 58 (1982), 9–12.
Morozov, A. Yu.: Integrability and matrix models, Uspekhi Fiz. Nauk 164 (1994), 3–62.
Orlov, A. Yu.: Vertex operators, ∂ bar problem, symmetries, Hamiltonian and Lagrangian formalism of (2+1) dimensional integrable systems, In: V. G. Bar’yakhtar and V. E. Zakharov (eds), Plasma Theory and Nonlinear and Turbulent Processes in Physics, Proc. III Kiev Intern. Workshop, Vol. I. Proceedings (Kiev 1987), World Scientific, Singapore, 1988, pp. 116–134.
Orlov, A. Yu.: Hypergeometric functions related to Schur Q-polynomials and BKP equation, to be published in Teor. Mat. Phys. 137(2) (November 2003).
Orlov, A. Yu. and Scherbin, D. M.: Fermionic representation for basic hypergeometric functions related to Schur polynomials, nlin.SI/0001001.
Orlov, A. Yu. and Winternitz, P.: Theoret. Math. Phys. 113 (1997), 1393–1417.
Orlov, A. Yu. and Scherbin, M. D.: Milne’s hypergeometric functions in terms of free fermions, J. Phys. A: Math. Gen. 34 (2001), 2295–2310.
Orlov, A. Yu. and Scherbin, D. M.: Multivariate hypergeometric functions as tau functions of Toda lattice and Kadomtsev–Petviashvili equation, Physica D 152–153 (March 2001), 51–56.
Orlov, A. Yu. and Scherbin, D. M.: Hypergeometric solutions of soliton equations, Teor. Mat. Phys. 128(1) (2001), 84–108.
Takasaki, K.: Initial value problem for the Toda lattice hierarchy, Adv. Stud. Pure Math. 4 (1984), 139–163.
Takasaki, K.: The Toda lattice hierarchy and generalized String equation, Comm. Math. Phys. 181 (1996), 131; hep-th/9506089, June 1995.
Takebe, T.: Representation theoretical meaning of initial value problem for the Toda lattice hierarchy I, LMP 21 (1991), 77–84.
Takebe, T.: Representation theoretical meaning of initial value problem for the Toda lattice hierarchy II, Publ. RIMS, Kyoto Univ. 27 (1991), 491–503.
Takebe, T. and Takasaki, K.: Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), 743–808; hep-th/94050096.
Ueno, K. and Takasaki, K.: Adv. Stud. Pure Math. 4 (1984), 1–95.
Vilenkin, N. Ya. and Klimyk, A. U.: Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions, Kluwer Academic Publishers, 1992.
Zabrodin, A., Kharchev, S., Mironov, A., Marshakov, A. and Orlov, A.: Nuclear Phys. B (1992).
Zakharov, V. E. and Shabat, A. B.: J. Funct. Anal. Appl. 8 (1974), 226, 13 (1979), 166.
Zakharov, V. E., Manakov, S. V., Novikov, S. P. and Pitaevsky, L. P.: The Theory of Solitons. The Inverse Scattering Method.
Zinn-Justion, P.: HCIZ integral and 2D Toda lattice hierarchy, math-ph/0202045.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Orlov, A.Y. Soliton Theory, Symmetric Functions and Matrix Integrals. Acta Appl Math 86, 131–158 (2005). https://doi.org/10.1007/s10440-005-0467-z
Issue Date:
DOI: https://doi.org/10.1007/s10440-005-0467-z