{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18502",
  "author_total": 1,
  "authors": [
    {
      "name": "Leonid Petrov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d0tfgc2f018n5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18502",
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  "id": "d0tfgc2f018n5",
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  "landing_url": "https://arxiv.org/abs/2609.18502",
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      "math-ph"
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      "openalex",
      "2607|Advanced Combinatorial Mathematics"
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  "pdf_url": "https://arxiv.org/pdf/2609.18502",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d0tfgc2f018n5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:01Z",
      "native_id": "2609.18502",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18502"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "From Yang-Baxter to Robinson-Schensted-Knuth",
  "title_text": "From Yang-Baxter to Robinson-Schensted-Knuth",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/d0tfgc2f018n5",
  "venue": "arXiv",
  "versions": []
}