{
  "abstract": "Homogeneous coordinate rings of Grassmannians are among the most fundamental instances of cluster algebras arising in ``nature.'' Despite their importance, however, the cluster variables of these cluster algebras are not well-understood, except in special cases. For the Grassmannian of 3-planes in n-space, Fomin and Pylyavskyy conjectured [FP16] that the cluster and frozen variables are exactly the indecomposable, non-elliptic web invariants which are arborizable. They proved that in particular, if an SL3 tensor diagram is a planar tree, its web invariant is a cluster or frozen variable. In this article, we give a new proof of their result for Gr(3, n) planar tree webs, and we generalize the result to Gr(4, n): if an SL4 tensor diagram for Gr(4, n) is a planar tree web, its web invariant is a cluster or frozen variable. Our proof uses some of the cluster quasihomomorphisms from [EZLP+23] and [EZPT+25] (namely, upper promotion and spurion promotion), as well as a new 6-leg pattern map.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38398",
  "author_total": 3,
  "authors": [
    {
      "name": "Jian-Rong Li"
    },
    {
      "name": "Chenglu Wang"
    },
    {
      "name": "Lauren Williams"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c12aw5era10fg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38398",
  "fulltext_url": null,
  "id": "c12aw5era10fg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38398",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7215096621",
  "pdf_url": "https://arxiv.org/pdf/2609.38398",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c12aw5era10fg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38398",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38398"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Grassmannian tree webs and cluster variables",
  "title_text": "Grassmannian tree webs and cluster variables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/c12aw5era10fg",
  "venue": "arXiv",
  "versions": []
}