{
  "abstract": "We introduce a diagram algebra $\\mathcal{K}_L$ based on Kuperberg's $A_2$-webs and prove a Schur--Weyl duality $\\mathcal{K}_L \\cong \\operatorname{End}_{U_q(\\mathfrak{sl}_3)}((V^+ \\oplus V^-)^{\\otimes L})$, where $V^\\pm$ are the fundamental representations of $U_q(\\mathfrak{sl}_3)$. Thus, $\\mathcal{K}_L$ is a natural $A_2$ analogue of the Temperley--Lieb algebra. We construct an explicit cellular structure for $\\mathcal{K}_L$, with the web basis as its cellular basis and cell modules indexed by dominant $U_q(\\mathfrak{sl}_3)$-weights. The bases of these modules are parametrized by dominant lattice paths via a recursive growth algorithm. Using this structure, we determine local generators of $\\mathcal{K}_L$ and establish web formulations of the first and second fundamental theorems of invariant theory. In particular, we show that the braid-group image, augmented by local idempotents, does not generate $\\mathcal{K}_L$, and identify an additional local generator. Finally, we derive a bimodule decomposition of $(V^+ \\oplus V^-)^{\\otimes L}$, yielding a complete set of irreducible $\\mathcal{K}_L$-modules for formal or generic $q$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18801",
  "author_total": 3,
  "authors": [
    {
      "name": "Azat M. Gainutdinov"
    },
    {
      "name": "Jesper Lykke Jacobsen"
    },
    {
      "name": "Yueheng Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7hx7ard571m92.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18801",
  "fulltext_url": null,
  "id": "7hx7ard571m92",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18801",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7221782880",
  "pdf_url": "https://arxiv.org/pdf/2610.18801",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7hx7ard571m92.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:11Z",
      "native_id": "2610.18801",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18801"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Cellular Algebra of $A_2$ webs",
  "title_text": "Cellular Algebra of A_2 webs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/7hx7ard571m92",
  "venue": "arXiv",
  "versions": []
}