{
  "id": "2012.01627",
  "version": "v1",
  "published": "2020-12-03T01:32:15.000Z",
  "updated": "2020-12-03T01:32:15.000Z",
  "title": "A combinatorial formula for the nabla operator",
  "authors": [
    "Erik Carlsson",
    "Anton Mellit"
  ],
  "comment": "35 Pages",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We present an LLT-type formula for a general power of the nabla operator applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $\\nabla^k e_n$, and the Elias-Hogancamp formula for $(\\nabla^k p_1^n,e_n)$ as corollaries. We give a direct proof of the theorem by verifying that the LLT expansion satisfies the defining properties of $\\nabla^k$, such as triangularity in the dominance order, as well as a geometric proof based on a method for counting bundles on $\\mathbb{P}^1$ due to the second author. These formulas are related to an affine paving of the type A unramified affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and also to Stanley's chromatic symmetric functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-03T01:32:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "20C30",
      "33D52",
      "05A30",
      "14M15",
      "14C05"
    ],
    "keywords": [
      "nabla operator",
      "combinatorial formula",
      "stanleys chromatic symmetric functions",
      "unramified affine springer fiber",
      "llt expansion satisfies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}