{
  "id": "2301.07221",
  "version": "v1",
  "published": "2023-01-17T23:10:52.000Z",
  "updated": "2023-01-17T23:10:52.000Z",
  "title": "On the Combinatorics of $\\mathbb{F}_1$-Representations of Pseudotree Quivers",
  "authors": [
    "Jaiung Jun",
    "Jaehoon Kim",
    "Alex Sistko"
  ],
  "comment": "26 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We investigate quiver representations over $\\mathbb{F}_1$. Coefficient quivers are combinatorial gadgets equivalent to $\\mathbb{F}_1$-representations of quivers. We focus on the case when the quiver $Q$ is a pseudotree. For such quivers, we first use the notion of coefficient quivers to provide a complete classification of asymptotic behaviors of indecomposable representations over $\\mathbb{F}_1$. Then, we prove some fundamental structural results about the Lie algebras associated to pseudotrees. Finally, we construct examples of $\\mathbb{F}_1$-representations $M$ of a quiver $Q$ by using coverings, under which the Euler characteristics of the quiver Grassmannians $\\textrm{Gr}^Q_{\\underline{d}}(M)$ can be computed in a purely combinatorial way.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-17T23:10:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "05E10",
      "16G60",
      "17B35"
    ],
    "keywords": [
      "pseudotree quivers",
      "combinatorics",
      "coefficient quivers",
      "combinatorial gadgets equivalent",
      "fundamental structural results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}