{
  "abstract": "Representations of quivers over the virtual field $\\mathbb{F}_1$ can be naturally extended to bound quivers with monomial and commutativity relations. In this paper, we introduce and study the Hall algebra $\\mathcal{H}_{\\mathbb{F}_1}(Q,I)$ for a bound quiver $(Q,I)$ equipped with such relations. As in the quiver setting, $\\mathcal{H}_{\\mathbb{F}_1}(Q,I)$ is a graded, connected, and cocommutative bialgebra, and the free subgroup generated by all indecomposable representations forms a Lie subalgebra. We study $\\mathcal{H}_{\\mathbb{F}_1}(Q,I)$ in detail for two special classes of bound quivers: Nakayama bound quivers and gentle one-cycle bound quivers. In both cases, we prove that for any triple $L, M, N$ of $\\mathbb{F}_1$-representations over $(Q,I)$, the Hall number $g_{N,L}^M$ coincides with the evaluation of the corresponding Hall polynomial $ϕ_{N,L}^M(t)$ at $t=1$. Consequently, there exists a canonical isomorphism between $\\mathcal{H}_{\\mathbb{F}_1}(Q,I)$ and the degenerate Hall algebra $\\mathcal{H}_1(Q,I)$, justifying Tits's $\\mathbb{F}_1$-philosophy in these settings.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02904",
  "author_total": 2,
  "authors": [
    {
      "name": "Changjian Fu"
    },
    {
      "name": "Xuelian Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8hzs98dmqmhq4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02904",
  "fulltext_url": null,
  "id": "8hzs98dmqmhq4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02904",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8hzs98dmqmhq4.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.02904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Representations of Bound Quivers over $\\mathbb{F}_1$ and Hall Algebras",
  "title_text": "Representations of Bound Quivers over 𝔽_1 and Hall Algebras",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/8hzs98dmqmhq4",
  "venue": "arXiv",
  "versions": []
}