A one-parameter refinement of the Razumov-Stroganov correspondence

TL;DR

This paper introduces a one-parameter refinement of the Razumov-Stroganov correspondence for fully-packed loop configurations, connecting enumeration vectors to the ground state of a deformed scattering matrix, generalizing previous results.

Contribution

It provides the first one-parameter refinement of the Razumov-Stroganov correspondence for generalized domains with gyration, linking enumeration vectors to an integrable deformation of the loop model Hamiltonian.

Findings

Enumeration vectors match the ground state of the deformed scattering matrix.

The refinement generalizes the original Razumov-Stroganov correspondence.

It confirms a conjecture by Di Francesco from 2004.

Abstract

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 generalize 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 to 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, follow from our results.