{
  "id": "1812.08345",
  "version": "v1",
  "published": "2018-12-20T03:14:45.000Z",
  "updated": "2018-12-20T03:14:45.000Z",
  "title": "Minuscule reverse plane partitions via quiver representations",
  "authors": [
    "Alexander Garver",
    "Rebecca Patrias",
    "Hugh Thomas"
  ],
  "comment": "Comments welcome",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "A nilpotent endomorphism of a quiver representation induces a linear transformation on the vector space at each vertex. Generically among all nilpotent endomorphisms, there is a well-defined Jordan form for these linear transformations, which is an interesting new invariant of a quiver representation. If $Q$ is a Dynkin quiver and $m$ is a minuscule vertex, we show that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the category of which we denote by $\\mathcal C_{Q,m}$, are determined up to isomorphism by this invariant. We use this invariant to define a bijection from isomorphism classes of representations in $\\mathcal C_{Q,m}$ to reverse plane partitions whose shape is the minuscule poset corresponding to $Q$ and $m$. By relating the piecewise-linear promotion action on reverse plane partitions to Auslander-Reiten translation in the derived category, we give a uniform proof that the order of promotion equals the Coxeter number. In type $A_n$, we show that special cases of our bijection include the Robinson-Schensted-Knuth and Hillman-Grassl correspondences.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-20T03:14:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20",
      "05E10"
    ],
    "keywords": [
      "minuscule reverse plane partitions",
      "linear transformation",
      "nilpotent endomorphism",
      "quiver representation induces",
      "piecewise-linear promotion action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}