{
  "abstract": "Two combinatorial rules are known for the Grassmannian Grothendieck polynomial $G^{(β)}_λ$: a sum over set-valued tableaux of shape $λ$, due to Buch, and a sum over the efficient cells of a cellular decomposition of the Gelfand-Zetlin polytope $GZ(λ)$, due to E. Presnova and the author. All coefficients in both sums equal $1$. We construct an explicit bijection between the two indexing sets which matches the summands term by term, carrying the number of excess entries of a tableau to the dimension of the corresponding cell; in particular the two rules are equivalent, either being deducible from the other. The efficiency condition on cells turns out to be the column-strictness of tableaux. We then transport Yu's square-root crystal operators to the cells and find that they respect dimension, along a double $i$-string the cells alternate between two consecutive dimensions, but not incidence: consecutive cells of such a string need not share a point, already for $λ=(2,1,0)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08760",
  "author_total": 1,
  "authors": [
    {
      "name": "Evgeny Smirnov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33kkbkk20mdcz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08760",
  "fulltext_url": null,
  "id": "33kkbkk20mdcz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08760",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212058965",
  "pdf_url": "https://arxiv.org/pdf/2609.08760",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33kkbkk20mdcz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.08760",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08760"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Set-valued tableaux and cells of Gelfand-Zetlin polytopes",
  "title_text": "Set-valued tableaux and cells of Gelfand-Zetlin polytopes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/33kkbkk20mdcz",
  "venue": "arXiv",
  "versions": []
}