{
  "id": "1707.01172",
  "version": "v1",
  "published": "2017-07-04T22:57:31.000Z",
  "updated": "2017-07-04T22:57:31.000Z",
  "title": "Polynomial bases: positivity and Schur multiplication",
  "authors": [
    "Dominic Searles"
  ],
  "comment": "23 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions, and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters and Demazure atoms; the quasi-key, fundamental and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial. This rule simultaneously extends Haglund, Luoto, Mason and van Willigenburg's (2011) Littlewood-Richardson rule for quasi-Schur polynomials and refines their Littlewood-Richardson rule for Demazure characters. We also establish bijections connecting combinatorial models for these polynomials including semi-skyline fillings and quasi-key tableaux.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-04T22:57:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10"
    ],
    "keywords": [
      "polynomial bases",
      "schur multiplication",
      "littlewood-richardson rule",
      "demazure characters",
      "schur polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}