{
  "id": "2008.11304",
  "version": "v1",
  "published": "2020-08-25T23:13:11.000Z",
  "updated": "2020-08-25T23:13:11.000Z",
  "title": "On quiver representations over $\\mathbb{F}_1$",
  "authors": [
    "Jaiung Jun",
    "Alex Sistko"
  ],
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We study the category $\\textrm{Rep}(Q,\\mathbb{F}_1)$ of representations of a quiver $Q$ over \"the field with one element\", denoted by $\\mathbb{F}_1$, and the Hall algebra of $\\textrm{Rep}(Q,\\mathbb{F}_1)$. Representations of $Q$ over $\\mathbb{F}_1$ often reflect combinatorics of those over $\\mathbb{F}_q$, but show some subtleties - for example, we prove that a connected quiver $Q$ is of finite type over $\\mathbb{F}_1$ if and only if $Q$ is a tree. Then, to each representation $\\mathbb{V}$ of $Q$ over $\\mathbb{F}_1$, we associate a combinatorial gadget $\\Gamma_\\mathbb{V}$, which we call a colored quiver, possessing the same information as $\\mathbb{V}$. This allows us to translate representations over $\\mathbb{F}_1$ purely in terms of combinatorics of associated colored quivers. We also explore the growth of indecomposable representations of $Q$ over $\\mathbb{F}_1$ searching for the tame-wild dichotomy over $\\mathbb{F}_1$ - this also shows a similar tame-wild dichotomy over a field, but with some subtle differences. Finally, we link the Hall algebra of the category of nilpotent representations of an $n$-loop quiver over $\\mathbb{F}_1$ with the Hopf algebra of skew shapes introduced by Szczesny.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-25T23:13:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "05E10",
      "16G60",
      "17B35"
    ],
    "keywords": [
      "quiver representations",
      "hall algebra",
      "colored quiver",
      "similar tame-wild dichotomy",
      "translate representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}