The representation of Bethe wave functions of certain integrable models via Schur functions allows one to apply the well-developed theory of symmetric functions to the calculation of thermal correlation functions. The algebraic relations arising in the calculation of scalar products and correlation functions are based on the Binet–Cauchy formula for the Schur functions. We provide a combinatorial interpretation of the formula for the scalar products of Bethe state vectors in terms of nests of self-avoiding lattice paths constituting so-called watermelon configurations. The proposed interpretation is, in turn, related to the enumeration of boxed plane partitions. Bibliography: 23 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, Oxford (1995).
W. Fulton, Young Tableaux with Application to Representation Theory and Geometry, Cambridge Univ. Press, Cambridge (1997).
D. M. Bressoud, Proofs and Confirmations. The Story of the Alternating Sign Matrix Conjecture, Cambridge Univ. Press, Cambridge (1999).
G. Schehr, S. N. Majumdar, A. Comtet, and P. J. Forrester, “Reunion probability of N vicious walkers: typical and large fluctuations for large N,” J. Stat. Phys., 149, 385–410 (2012).
P. Zinn-Justin, “Six-vertex model with domain wall boundary conditions and one-matrix model,” Phys. Rev. E, 62, 3411–3418 (2000).
A. Okounkov, “Symmetric functions and random partitions,” in: Symmetric Functions 2001: Surveys of Developments and Perspectives, NATO Science Series, Vol. 74 (2002), pp. 223–252.
K. Hikami and T. Imamura, “Vicious walkers and hook Young tableaux,” J. Phys. A: Math. Gen., 36, 3033–3048 (2003).
A. Okounkov and N. Reshetikhin, “Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram,” J. Amer. Math. Soc., 16, 581–603 (2003).
G. Téllez and P. J. Forrester, “Expanded Vandermonde powers and sum rules for the two-dimensional onecomponent plasma,” J. Stat. Phys., 148, 824–855 (2012).
N. M. Bogoliubov, “Boxed plane partitions as an exactly solvable boson model,” J. Phys. A: Math. Gen., 38, 9415–9430 (2005).
N. M. Bogoliubov and J. Timonen, “Correlation functions for a strongly coupled boson system and plane partitions,” Phil. Trans. Roy. Soc. A, 369, 1319–1333 (2011).
N. M. Bogoliubov, “XX0 Heisenberg chain and random walks,” J. Math. Sci., 138, 5636–5643 (2006).
N. M. Bogoliubov, “The integrable models for the vicious and friendly walkers,” J. Math. Sci., 143, 2729–2737 (2007).
N. M. Bogoliubov and C. Malyshev, “The correlation functions of the XX Heisenberg magnet and random walks of vicious walkers,” Theor. Math. Phys., 159, 563–574 (2009).
N. M. Bogoliubov and C. Malyshev, “The correlation functions of the XXZ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicioius walkers,” St.Petersburg Math. J., 22, 359–377 (2011).
N. M. Bogoliubov and C. Malyshev, “Correlation functions of the XXZ chain at zero anisotropy and enumeration of boxed plane partitions,” PDMI Preprint 19/2012 (2012).
L. D. Faddeev, “Quantum completely integrable models of field theory,” Sov. Sci. Rev. Math. C, 1, 107–160 (1980).
V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Univ. Press, Cambridge (1993).
G. Kuperberg, “Another proof of the alternating sign matrix conjecture,” Int. Math. Res. Notices, 1996, 139–150 (1996).
A. Klimyk and K. Schmudgen, Quantum Groups and their Representations, Springer, Berlin (1997).
I. Gessel and X. G. Viennot, “Determinants, paths, and plane partitions,” preprint (1989).
I. Gessel and G. Viennot, “Binomial determinants, paths, and hook length formulas,” Adv. Math., 58, 300–321 (1985).
A. J. Guttmann, A. L. Owczarek, and X. G. Viennot, “Vicious walkers and Young tableaux. I: Without walls,” J. Phys. A: Math. Gen., 31, 8123–8135 (1998).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 421, 2014, pp. 33–46.
Rights and permissions
About this article
Cite this article
Bogoliubov, N.M., Malyshev, C. A Combinatorial Interpretation of the Scalar Products of State Vectors of Integrable Models. J Math Sci 200, 662–670 (2014). https://doi.org/10.1007/s10958-014-1956-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1956-2