{
  "id": "1609.04542",
  "version": "v1",
  "published": "2016-09-15T09:20:32.000Z",
  "updated": "2016-09-15T09:20:32.000Z",
  "title": "Decomposition rules for the ring of representations of non-Archimedean $GL_n$",
  "authors": [
    "Maxim Gurevich"
  ],
  "comment": "32 pages, contains the results of the previous note arXiv:1604.07333",
  "categories": [
    "math.RT",
    "math.QA"
  ],
  "abstract": "Let $\\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the groups $\\{GL_n(F)\\}_{n=0}^\\infty$ ($F$ a fixed $p$-adic field) taken together, with multiplication defined in the sense of parabolic induction. We study the problem of the decomposition of a product of irreducible representations in $\\mathcal{R}$. We introduce a width invariant for elements of the ring. By showing that the invariant gives an increasing ring filtration, we obtain a necessary condition on irreducible factors of a given product. Irreducible representations of width $1$ form the previously studied class of ladder representations. We later focus on the case of a product of two ladder representations, for which we establish that all irreducible factors appear with multiplicity one. Finally, we conjecture a general rule for the composition series of a product of two ladder representations and prove its validity for cases in which the irreducible factors correspond to smooth Schubert varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-15T09:20:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G25",
      "22E50"
    ],
    "keywords": [
      "decomposition rules",
      "ladder representations",
      "irreducible factors",
      "non-archimedean",
      "complex smooth finite-length representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}