{
  "abstract": "We introduce posets of type $\\mathbb{D}$, a family of posets described by an admissible Dynkin quiver of type $D_n$ together with a compatible set of extra arrows (alien arrows), and show that their socle-projective representation category is of finite representation type. Continuing, in the same spirit, a program initiated for posets of type $\\mathbb{A}$ [R. Schiffler, R-J. Serna, A geometric realization of socle-projective categories for posets of type $\\mathbb{A}$, J. Pure Appl. Algebra 224 (2020), no. 12, 106436], we build on an existing geometric model for cluster-tilted algebras of type $D_n$, whose geometric combinatorics is more intricate than that of type $\\mathbb{A}$. Our main result establishes a $\\Bbbk$-linear categorical equivalence $Θ$ between a full subcategory $(\\mathcal C/T)_F$ of the arc category of a punctured $(n+3)$-gon, whose objects are certain $sp$-arcs, and the category of finitely generated socle-projective $\\Bbbk\\mathscr{P}$-modules, for $\\mathscr{P}$ a poset of type $\\mathbb{D}$. As a consequence, when the set of alien arrows is empty, we conclude that the cluster subalgebra generated by the $sp$-arcs coincides with the full cluster algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25174",
  "author_total": 3,
  "authors": [
    {
      "name": "Gabriel Bravo Rios"
    },
    {
      "name": "Ralf Schiffler"
    },
    {
      "name": "Robinson-Julian Serna"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1bawvj6mm3nhh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25174",
  "fulltext_url": null,
  "id": "1bawvj6mm3nhh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25174",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7214084988",
  "pdf_url": "https://arxiv.org/pdf/2609.25174",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1bawvj6mm3nhh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25174",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25174"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A geometric realization of socle-projective categories for posets of type $\\mathbb{D}$",
  "title_text": "A geometric realization of socle-projective categories for posets of type 𝔻",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/1bawvj6mm3nhh",
  "venue": "arXiv",
  "versions": []
}