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

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.