From Yang-Baxter to Robinson-Schensted-Knuth

2026/09/16

We explain how to derive the Robinson-Schensted-Knuth (RSK) correspondence, a fundamental bijection in algebraic combinatorics, from the Yang-Baxter equation. The Yang-Baxter equation arose in the study of quantum many-body systems and later became a cornerstone of the theory of solvable lattice models, particularly vertex models. In a vertex model, arrows occupy the edges of a grid, and each vertex carries a Boltzmann weight determined by the arrows on the four edges meeting at it. The weight of a configuration is the product of these local weights, and a partition function is the sum of the weights of all configurations with prescribed boundary conditions. For a vertex model whose partition functions are the Schur polynomials, the two sides of each instance of the Yang-Baxter equation admit exactly one weight-preserving matching of their summands. Carried across a grid, this forced matching is the classical RSK correspondence in the form of Fomin’s growth diagrams. In natural coordinates the local matching becomes the combinatorial three-dimensional $R$, a set-theoretic solution of the Zamolodchikov tetrahedron equation. Its periodic closure returns the combinatorial $R$-matrices of one-row crystals.

The forced matching is special to the Schur weights. For the Hall-Littlewood and $q$-Whittaker deformations and their spin versions, at generic parameter values no deterministic matching works for all boundary data. Reading each instance of the Yang-Baxter equation probabilistically, we replace the matching by a coupling of the two sides — a bijectivization, or probabilistic bijection — and obtain Markov operators that transport probability measures attached to vertex models. Iterated over the grid, these operators produce randomized RSK-type dynamics and interacting particle systems, including $q$-PushTASEP and the stochastic six-vertex model.

This survey grew out of lectures given at Institut Mittag-Leffler in the summer of 2026.