A one-parameter refinement of the Razumov–Stroganov correspondence

We introduce and prove a one-parameter refinement of the Razumov–Stroganov correspondence. This is achieved for fully-packed loop configurations (FPL) on domains which generalise the square domain, and which are endowed with the gyration operation. We consider one given side of the domain, and FPLs such that the only straight-line tile on this side is black. We show that the enumeration vector associated with such FPLs, weighted according to the position of the straight line and refined according to the link pattern for the black boundary points, is the ground state of the scattering matrix, an integrable one-parameter deformation of the O ( 1 ) Dense Loop Model Hamiltonian. We show how the original Razumov–Stroganov correspondence, and a conjecture formulated by Di Francesco in 2004, follows from our results.