{
  "id": "2403.04597",
  "version": "v1",
  "published": "2024-03-07T15:45:19.000Z",
  "updated": "2024-03-07T15:45:19.000Z",
  "title": "Scalar extensions of quiver representations over $\\mathbb{F}_1$",
  "authors": [
    "Markus Kleinau"
  ],
  "comment": "16 pages, comments welcome",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Let $V$ and $W$ be quiver representations over $\\mathbb{F}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of $\\mathrm{Hom}_{KQ}(V^K,W^K)$ generalising the well-known basis of the morphism spaces between string and tree modules. We use this basis to give a combinatorial characterisation of absolutely indecomposable representations. Furthermore, we show that indecomposable representations with finite nice length are absolutely indecomposable. This answers a question of Jun and Sistko.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-07T15:45:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quiver representations",
      "scalar extensions",
      "finite nice length",
      "well-known basis",
      "combinatorial characterisation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}