{
  "abstract": "Using Galois covering theory and equivariant techniques, we classify all connected representation-tame Geiß--Leclerc--Schröer (GLS) algebras in terms of their defining triples $(C,D,Ω)$. We then study the GLS algebras $H(\\widetilde{CD}_n)$ with minimal symmetrizers, where $n\\geq 2$ and $\\widetilde{CD}_2=\\widetilde{B}_2$. By realizing these algebras as basic algebras of $\\mathbb{Z}_2$-skew group algebras of the string algebras $H(\\widetilde{C}_{2n-2})$, we introduce extended strings and extended bands to parameterize the connected components of the Auslander--Reiten quivers of $H(\\widetilde{CD}_n)$ and determine their shapes. We further classify the indecomposable $τ$-locally free modules over representation-tame GLS algebras of affine type and show that their rank vectors form precisely the set of positive roots together with explicitly described non-root vectors in the positive cone of the root lattice. This yields a precise revision of the GLS conjecture for representation-tame GLS algebras of affine type.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26503",
  "author_total": 4,
  "authors": [
    {
      "name": "Qiang Dong"
    },
    {
      "name": "Zengqiang Lin"
    },
    {
      "name": "Ming Lu"
    },
    {
      "name": "Shiquan Ruan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ejm3tz1y17yfr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26503",
  "fulltext_url": null,
  "id": "ejm3tz1y17yfr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26503",
  "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.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7214066660",
  "pdf_url": "https://arxiv.org/pdf/2609.26503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ejm3tz1y17yfr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26503"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Representation-tame Geiß-Leclerc-Schröer algebras and a revised GLS conjecture on root systems",
  "title_text": "Representation-tame Geiß-Leclerc-Schröer algebras and a revised GLS conjecture on root systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/ejm3tz1y17yfr",
  "venue": "arXiv",
  "versions": []
}