{
  "id": "2009.06427",
  "version": "v1",
  "published": "2020-09-14T13:25:05.000Z",
  "updated": "2020-09-14T13:25:05.000Z",
  "title": "Poles of finite-dimensional representations of Yangians in type A",
  "authors": [
    "Sachin Gautam",
    "Curtis Wendlandt"
  ],
  "comment": "27 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\\mathbb{C}$, and let $Y_{\\hbar}(\\mathfrak{g})$ be the Yangian of $\\mathfrak{g}$. In this paper, we initiate the study of the set of poles of the rational currents defining the action of $Y_{\\hbar}(\\mathfrak{g})$ on an arbitrary finite-dimensional vector space $V$. We prove that this set is completely determined by the eigenvalues of the commuting Cartan currents of $Y_{\\hbar}(\\mathfrak{g})$, and therefore encodes the singularities of the components of the $q$-character of $V$. In type $\\mathsf{A}$, we explicitly determine the set of poles of every irreducible $V$ in terms of the roots of the underlying Drinfeld polynomials. In particular, our results yield a complete classification of the finite-dimensional irreducible representations of the Yangian double in type $\\mathsf{A}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-14T13:25:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B37",
      "81R10"
    ],
    "keywords": [
      "finite-dimensional representations",
      "finite-dimensional simple lie algebra",
      "arbitrary finite-dimensional vector space",
      "results yield",
      "drinfeld polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}