{
  "id": "2112.06291",
  "version": "v2",
  "published": "2021-12-12T18:15:51.000Z",
  "updated": "2023-12-13T19:28:36.000Z",
  "title": "Coefficient Quivers, $\\mathbb{F}_1$-Representations, and Euler Characteristics of Quiver Grassmannians",
  "authors": [
    "Jaiung Jun",
    "Alex Sistko"
  ],
  "comment": "50 pages, majors revisions to exposition, new appendix. To appear in Nagoya Math. J",
  "doi": "10.1017/nmj.2023.37",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "A quiver representation assigns a vector space to each vertex, and a linear map to each arrow. When one considers the category $\\textrm{Vect}(\\mathbb{F}_1)$ of vector spaces ``over $\\mathbb{F}_1$'' (the field with one element), one obtains $\\mathbb{F}_1$-representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category $\\textrm{Rep}(Q,\\mathbb{F}_1)$ is equivalent to the (suitably defined) category of coefficient quivers over $Q$. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of ``$\\mathbb{F}_1$-rational points'' of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated to $\\mathbb{F}_1$-representations. These techniques apply to a large class of $\\mathbb{F}_1$-representations, which we call the $\\mathbb{F}_1$-representations with finite nice length: we prove sufficient conditions for an $\\mathbb{F}_1$-representation to have finite nice length, and classify such representations for certain families of quivers. Finally, we explore the Hall algebras associated to $\\mathbb{F}_1$-representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent $\\mathbb{F}_1$-representations of a quiver with bounded representation type. We also discuss Hall algebras associated to representations with finite nice length, and compute them for certain families of quivers.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-12-13T19:28:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "05E10",
      "16T30",
      "17B35"
    ],
    "keywords": [
      "quiver grassmannians",
      "euler characteristics",
      "coefficient quivers",
      "finite nice length",
      "hall algebra"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}