Generalizations of the Razumov–Stroganov correspondence

Establish generalizations of the Razumov–Stroganov correspondence asserting that, for six-vertex model configurations with Domain Wall Boundary Conditions mapped to Fully Packed Loop configurations, the vector of configuration counts refined by boundary connectivity is an eigenvector of the Hamiltonian of a quantum integrable system, beyond the case proven by Cantini and Sportiello.

Background

In the Razumov–Stroganov correspondence, configurations of the six-vertex model with Domain Wall Boundary Conditions are mapped to Fully Packed Loop (FPL) configurations, and one performs a refined counting by the connectivity pattern of boundary edges. Razumov and Stroganov conjectured that the resulting counting vector is an eigenvector of a matrix which is identified with the Hamiltonian of a quantum integrable system.

This conjecture was proven by Cantini and Sportiello using the gyration operation introduced by Wieland, but the authors note that various generalizations remain open. The generalizations concern extensions of the correspondence to broader contexts in which similar eigenvector relationships between refined FPL enumerations and integrable Hamiltonians are expected.