{
  "id": "2309.04753",
  "version": "v1",
  "published": "2023-09-09T10:52:19.000Z",
  "updated": "2023-09-09T10:52:19.000Z",
  "title": "On the Spectrum of Exterior Algebra, and Generalized Exponents of Small Representations",
  "authors": [
    "Sabino Di Trani"
  ],
  "comment": "28 pages. Comments are welcome. To appear in Bollettino dell'Unione Matematica Italiana",
  "categories": [
    "math.RT",
    "math.CO",
    "math.RA"
  ],
  "abstract": "We present some results about the irreducible representations appearing in the exterior algebra $\\Lambda \\mathfrak{g}$, where $ \\mathfrak{g}$ is a simple Lie algebra over $\\mathbb{C}$. For Lie algebras of type $B$, $C$ or $D$ we prove that certain irreducible representations, associated to weights characterized in a combinatorial way, appear as irreducible components of $\\Lambda \\mathfrak{g}$. Moreover, we propose an analogue of a conjecture of Kostant, about irreducibles appearing in the exterior algebra of the little adjoint representation. Finally, we give some closed expressions, in type $B$, $C$ and $D$, for generalized exponents of small representations that are fundamental representations and we propose a generalization of some results of De Concini, M\\\"oseneder Frajria, Procesi and Papi about the module of special covariants of adjoint and little adjoint type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-09T10:52:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exterior algebra",
      "small representations",
      "generalized exponents",
      "little adjoint type",
      "irreducible representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}