{
  "id": "1907.07616",
  "version": "v1",
  "published": "2019-07-17T16:17:26.000Z",
  "updated": "2019-07-17T16:17:26.000Z",
  "title": "Plethysms of symmetric functions and representations of $\\mathrm{SL}_2(\\mathbb{C})$",
  "authors": [
    "Rowena Paget",
    "Mark Wildon"
  ],
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Let $\\nabla^\\lambda$ denote the Schur functor labelled by the partition $\\lambda$ and let $E$ be the natural representation of $\\mathrm{SL}_2(\\mathbb{C})$. We make a systematic study of when there is an isomorphism $\\nabla^\\lambda \\!\\mathrm{Sym}^\\ell \\!E \\cong \\nabla^\\mu \\!\\mathrm{Sym}^m \\! E$ of representations of $\\mathrm{SL}_2(\\mathbb{C})$. Generalizing earlier results of King and Manivel, we classify all such isomorphisms when $\\lambda$ and $\\mu$ are conjugate partitions and when one of $\\lambda$ or $\\mu$ is a rectangle. We give a complete classification when $\\lambda$ and $\\mu$ each have at most two rows or columns or is a hook partition and a partial classification when $\\ell = m$. As a corollary of a more general result on Schur functors labelled by skew partitions we also determine all cases when $\\nabla^\\lambda \\!\\mathrm{Sym}^\\ell \\!E$ is irreducible. The methods used are from representation theory and combinatorics; in particular, we make explicit the close connection with MacMahon's enumeration of plane partitions, and prove a new $q$-binomial identity in this setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-17T16:17:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric functions",
      "schur functor",
      "natural representation",
      "binomial identity",
      "systematic study"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}