Abstract
Studying a possible connection between the ground-state vector for some special spin systems and the so-called alternating-sign matrices, we find numerical evidence that the components of the ground-state vector of the O(1) loop model coincide with the numbers of the states of the so-called fully packed loop model with fixed pairing patterns. The states of the latter system are in one-to-one correspondence with alternating-sign matrices. This allows advancing the hypothesis that the components of the ground-state vector of the O(1) loop model coincide with the cardinalities of the corresponding subsets of the alternating-sign matrices. In a sense, our conjecture generalizes the conjecture of Bosley and Fidkowski, which was refined by Cohn and Propp and proved by Wieland.
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REFERENCES
A. V. Razumov and Yu. G. Stroganov, J. Phys. A, 34, 3185–3190 (2001); cond-mat/0012141.
M. T. Batchelor, J. de Gier, and B. Nienhuis, J. Phys. A, 34, L265–L270 (2001); cond-mat/0101385.
A. V. Razumov and Yu. G. Stroganov, J. Phys. A, 34, 5335–5340 (2001); cond-mat/0102247.
H. W. J. Blöte, and B. Nienhuis, J. Phys. A, 22, 1415–1438 (1989).
V. E. Korepin, A. G. Izergin, and N. M. Bogoliubov, Quantum Inverse Scattering Method, Correlation Functions, and Algebraic Bethe Ansatz, Cambridge Univ. Press, Cambridge (1993).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, London (1982).
J. Propp, Discrete Math. Theor. Comput. Sci. (Proc.), AA, 43–58 (2001); http://dmtcs.loria.fr/proceedings/ html/dmAA0103.abs.html.
B. Wieland, Electron. J. Combin., 7, No. 1, R37 (2000); www.combinatorics.org.
P. Lankaster, Theory of Matrices, Acad. Press, New York (1969).
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Razumov, A.V., Stroganov, Y.G. Combinatorial Nature of the Ground-State Vector of the O(1) Loop Model. Theoretical and Mathematical Physics 138, 333–337 (2004). https://doi.org/10.1023/B:TAMP.0000018450.36514.d7
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DOI: https://doi.org/10.1023/B:TAMP.0000018450.36514.d7