Combinatorial Nature of the Ground-State Vector of the O(1) Loop Model

By A. Razumov, Y. Stroganov. 2001

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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.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/85f84cab54d387967579d230b50e377a4ec496ca