{
  "id": "2212.09419",
  "version": "v1",
  "published": "2022-12-19T12:52:29.000Z",
  "updated": "2022-12-19T12:52:29.000Z",
  "title": "Toward Butler's conjecture",
  "authors": [
    "Donghyun Kim",
    "Seung Jin Lee",
    "Jaeseong Oh"
  ],
  "comment": "49 pages, 12 figures, 12 tables, comments are welcome",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "For a partition $\\nu$, let $\\lambda,\\mu\\subseteq \\nu$ be two distinct partitions such that $|\\nu/\\lambda|=|\\nu/\\mu|=1$. Butler conjectured that the divided difference $\\operatorname{I}_{\\lambda,\\mu}[X;q,t]=(T_\\lambda\\widetilde{H}_\\mu[X;q,t]-T_\\mu\\widetilde{H}_\\lambda[X;q,t])/(T_\\lambda-T_\\mu)$ of modified Macdonald polynomials of two partitions $\\lambda$ and $\\mu$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for $\\operatorname{I}_{\\lambda,\\mu}[X;q,t]$, which is a positive monomial expansion. We also prove Butler's conjecture for some special cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-19T12:52:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05",
      "05E10",
      "05A05"
    ],
    "keywords": [
      "butlers conjecture",
      "column exchange rule",
      "llt equivalence",
      "distinct partitions",
      "modified macdonald polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}