O(1) loop model with different boundary conditions and symmetry classes of alternating-sign matrices

Abstract

This work as an extension of our recent paper where we have found a numerical evidence for the fact that the numbers of the states of the fully packed loop (FPL) model with fixed link-patterns coincide with the components of the ground state vector of the dense O(1) loop model for periodic boundary conditions and an even number of sites. Here we give two new conjectures related to different boundary conditions. Namely, we suggest that the numbers of the half-turn symmetric states of the FPL model with fixed link-patterns coincide with the components of the ground state vector of the dense O(1) loop model for periodic boundary conditions and an odd number of sites and that the corresponding numbers of the vertically symmetric states describe the case of the open boundary conditions and an even number of sites.