{
  "abstract": "Let $k$ be an algebraically closed field and let $Λ$ be a finite-dimensional associative $k$-algebra. We apply Joyce and Brideland's notion of Ringel--Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for $Λ$, i.e. a faithful functor from the $τ$-cluster morphism category of $Λ$ into a groupoid. Consequently, by results of Hanson--Igusa, the picture space of $Λ$ is locally CAT(0) provided that the $τ$-cluster morphism category of $Λ$ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa--Todorov beyond hereditary $k$-algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29960",
  "author_total": 1,
  "authors": [
    {
      "name": "Erlend D. Børve"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ffrvbsktwgrzj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29960",
  "fulltext_url": null,
  "id": "ffrvbsktwgrzj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29960",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7204877252",
  "pdf_url": "https://arxiv.org/pdf/2608.29960",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ffrvbsktwgrzj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29960",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29960"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Picture groups and green sequences from the perspective of Ringel--Hall algebras",
  "title_text": "Picture groups and green sequences from the perspective of Ringel--Hall algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/ffrvbsktwgrzj",
  "venue": "arXiv",
  "versions": []
}