{
  "id": "2404.04512",
  "version": "v1",
  "published": "2024-04-06T05:57:53.000Z",
  "updated": "2024-04-06T05:57:53.000Z",
  "title": "From quasi-symmetric to Schur expansions with applications to symmetric chain decompositions and plethysm",
  "authors": [
    "Rosa Orellana",
    "Franco Saliola",
    "Anne Schilling",
    "Mike Zabrocki"
  ],
  "comment": "27 pages; 5 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Egge, Loehr and Warrington provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide the Schur expansion of $s_w[s_h](x,y)$ for $w=2,3,4$ using novel symmetric chain decompositions of Young's lattice for partitions in a $w\\times h$ box. For $w=4$, this is first known combinatorial expression for the coefficient of $s_{\\lambda}$ in $s_{w}[s_{h}]$ for two-row partitions $\\lambda$, and for $w=3$ the combinatorial expression is new.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-06T05:57:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10"
    ],
    "keywords": [
      "schur expansion",
      "application",
      "novel symmetric chain decompositions",
      "combinatorial expression",
      "fundamental expansion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}