{
  "id": "1507.07832",
  "version": "v1",
  "published": "2015-07-28T16:26:18.000Z",
  "updated": "2015-07-28T16:26:18.000Z",
  "title": "Matrix factorizations for domestic triangle singularities",
  "authors": [
    "Dawid Edmund Kędzierski",
    "Helmut Lenzing",
    "Hagen Meltzer"
  ],
  "journal": "Colloq. Math. 140 (2015), 239-278",
  "doi": "10.4064/cm140-2-6",
  "categories": [
    "math.RT"
  ],
  "abstract": "Working over an algebraically closed field $k$ of any characteristic, we determine the matrix factorizations for the --- suitably graded --- triangle singularities $f=x^a+y^b+z^c$ of domestic type, that is, we assume that $(a,b,c)$ are integers at least two, satisfying $1/a+1/b+1/c>1$. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type $(a,b,c)$. Equivalently, in a representation-theoretic context, we can work in the mesh category of $\\mathbb{Z}\\tilde\\Delta$ over $k$, where $\\tilde\\Delta$ is the extended Dynkin diagram, corresponding to the Dynkin diagram $\\Delta=[a,b,c]$. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the $\\mathbb{Z}$-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from $\\{0,\\pm1\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-28T16:26:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J17",
      "13H10",
      "16G60",
      "16G70"
    ],
    "keywords": [
      "domestic triangle singularities",
      "symmetric matrix factorizations",
      "extended dynkin diagram",
      "vector bundles",
      "domestic type"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}