Integrable combinatorics

What is the link between Alternating Sign Matrices (ASM), Totally Symmetric Self-Complementary Plane Partiitions (TSSCPP) and the equivariant cohomology of the variety of strictly upper triangular matrices with vanishing square?

Two-dimensional Integrable lattice models from statistical physics provide the natural framework for this missing link. We show in particular how a physical model, involving densely-packed loop configurations on an infinite surface, is connected to all three subjects above. This model turns out to be integrable, and we'll use this fact to reformulate all of the above in terms of polynomial solutions of the quantum Knizhnik-Zamolodchikov equation. Results include a proof of the Razumov-Stroganov sum rule, a new connection between ASM and TSSCPP, and the computation of the (multi)degree of the variety M=0. We also present generalizations to the commuting variety and to M^k=0.