{
  "id": "1711.10000",
  "version": "v1",
  "published": "2017-11-27T21:05:08.000Z",
  "updated": "2017-11-27T21:05:08.000Z",
  "title": "Necessary conditions for Schur-maximality",
  "authors": [
    "Foster Tom",
    "Stephanie van Willigenburg"
  ],
  "comment": "46 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that $B\\leq_s A$ if $s_A-s_B$ is Schur-positive. Towards this, McNamara and van Willigenburg proved that it suffices to study equitable ribbons, namely ribbons whose row lengths are all of length $a$ or $(a+1)$ for $a\\geq 2$. In this paper we confirm the conjecture of McNamara and Pylyavskyy in all cases where the comparable equitable ribbons form a chain. We also confirm a conjecture of McNamara and van Willigenburg regarding which equitable ribbons in general are minimal. Additionally, we establish two sufficient conditions for the difference of two ribbons to be Schur-positive, which manifest as diagrammatic operations on ribbons. We also deduce two necessary conditions for the difference of two equitable ribbons to be Schur-positive that rely on rows of length $a$ being at the end, or on rows of length $(a+1)$ being evenly distributed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-27T21:05:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10",
      "06A05",
      "06A06",
      "20C30"
    ],
    "keywords": [
      "necessary conditions",
      "van willigenburg",
      "schur-maximality",
      "difference",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}