{
  "id": "2306.03818",
  "version": "v1",
  "published": "2023-06-06T16:05:09.000Z",
  "updated": "2023-06-06T16:05:09.000Z",
  "title": "Non-degenerate potentials on the quiver $X_7$",
  "authors": [
    "Sefi Ladkani"
  ],
  "comment": "44 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "We develop a method to compute certain mutations of quivers with potentials and use this to construct an explicit family of non-degenerate potentials on the exceptional quiver $X_7$. We confirm a conjecture of Geiss-Labardini-Schroer by presenting a computer-assisted proof that over a ground field of characteristic 2, the Jacobian algebra of one member $W_0$ of this family is infinite-dimensional, whereas that of another member $W_1$ is finite-dimensional, implying that these potentials are not right equivalent. As a consequence, we draw some conclusions on the associated cluster categories, and in particular obtain a representation theoretic proof that there are no reddening mutation sequences for the quiver $X_7$. We also show that when the characteristic of the ground field differs from 2, the Jacobian algebras of $W_0$ and $W_1$ are both finite-dimensional. Thus $W_0$ seems to be the first known non-degenerate potential with the property that the finite-dimensionality of its Jacobian algebra depends upon the ground field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-06T16:05:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-degenerate potential",
      "jacobian algebra",
      "ground field differs",
      "representation theoretic proof",
      "reddening mutation sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}