{
  "id": "1704.08643",
  "version": "v1",
  "published": "2017-04-27T16:27:26.000Z",
  "updated": "2017-04-27T16:27:26.000Z",
  "title": "Factorization formulas of $K$-$k$-Schur functions I",
  "authors": [
    "Motoki Takigiku"
  ],
  "comment": "36 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We give some new formulas about factorizations of $K$-$k$-Schur functions $g^{(k)}_{\\lambda}$, analogous to the $k$-rectangle factorization formula $s^{(k)}_{R_t\\cup\\lambda}=s^{(k)}_{R_t}s^{(k)}_{\\lambda}$ of $k$-Schur functions, where $\\lambda$ is any $k$-bounded partition and $R_t$ denotes the partition $(t^{k+1-t})$ called \\textit{$k$-rectangle}. Although a formula of the same form does not hold for $K$-$k$-Schur functions, we can prove that $g^{(k)}_{R_t}$ divides $g^{(k)}_{R_t\\cup\\lambda}$, and in fact more generally that $g^{(k)}_{P}$ divides $g^{(k)}_{P\\cup\\lambda}$ for any multiple $k$-rectangles $P=R_{t_1}^{a_1}\\cup\\dots\\cup R_{t_m}^{a_m}$ and any $k$-bounded partition $\\lambda$. We give the factorization formula of such $g^{(k)}_{P}$ and the explicit formulas of $g^{(k)}_{P\\cup\\lambda}/g^{(k)}_{P}$ in some cases, including the case where $\\lambda$ is a partition with a single part as the easiest example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-27T16:27:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schur functions",
      "bounded partition",
      "rectangle factorization formula",
      "explicit formulas",
      "single part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}