{
  "id": "2310.03626",
  "version": "v1",
  "published": "2023-10-05T16:02:46.000Z",
  "updated": "2023-10-05T16:02:46.000Z",
  "title": "The cluster complex for cluster Poisson varieties and representations of acyclic quivers",
  "authors": [
    "Carolina Melo",
    "Alfredo Nájera Chávez"
  ],
  "comment": "20 pages, comments are welcome",
  "categories": [
    "math.RT",
    "math.AG",
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{X}$ be a skew-symmetrizable cluster Poisson variety. The cluster complex $\\Delta^+(\\mathcal{X})$ was introduced by Gross, Hacking, Keel and Kontsevich. It codifies the theta functions on $\\mathcal{X}$ that restrict to a character of a seed torus. Every seed ${ \\bf s}$ for $\\mathcal{X}$ determines a fan realization $\\Delta^+_{\\bf s}(\\mathcal{X})$ of $\\Delta^+(\\mathcal{X})$. For every ${\\bf s}$ we provide a simple and explicit description of the cones of $\\Delta^+_{{\\bf s}}(\\mathcal{X})$ and their facets using ${\\bf c}$-vectors. Moreover, we give formulas for the theta functions parametrized by the integral points of $\\Delta^+_{{ \\bf s}}(\\mathcal{X})$ in terms of $F$-polynomials. In case $\\mathcal{X}$ is skew-symmetric and the quiver $Q$ associated to ${\\bf s}$ is acyclic, we describe the normal vectors of the supporting hyperplanes of the cones of $\\Delta^+_{\\bf s}(\\mathcal{X})$ using ${\\bf g}$-vectors of (non-necessarily rigid) objects in $\\mathsf{K}^{\\rm b}(\\text{proj} \\; kQ)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-05T16:02:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60"
    ],
    "keywords": [
      "cluster complex",
      "acyclic quivers",
      "theta functions",
      "representations",
      "skew-symmetrizable cluster poisson variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}