{
  "id": "1006.1381",
  "version": "v1",
  "published": "2010-06-07T21:39:09.000Z",
  "updated": "2010-06-07T21:39:09.000Z",
  "title": "Categories parametrized by schemes and representation theory in complex rank",
  "authors": [
    "Akhil Mathew"
  ],
  "comment": "26 pages; Comments welcome",
  "categories": [
    "math.RT"
  ],
  "abstract": "Many key invariants in the representation theory of classical groups (symmetric groups $S_n$, matrix groups $GL_n$, $O_n$, $Sp_{2n}$) are polynomials in $n$ (e.g., dimensions of irreducible representations). This allowed Deligne to extend the representation theory of these groups to complex values of the rank $n$. Namely, Deligne defined generically semisimple families of tensor categories parametrized by $n\\in \\mathbb{C}$, which at positive integer $n$ specialize to the classical representation categories. Using Deligne's work, Etingof proposed a similar extrapolation for many non-semisimple representation categories built on representation categories of classical groups, e.g., degenerate affine Hecke algebras (dAHA). It is expected that for generic $n\\in \\mathbb{C}$ such extrapolations behave as they do for large integer $n$ (\"stabilization\"). The goal of our work is to provide a technique to prove such statements. Namely, we develop an algebro-geometric framework to study categories indexed by a parameter $n$, in which the set of values of $n$ for which the category has a given property is constructible. This implies that if a property holds for integer $n$, it then holds for generic complex $n$. We use this to give a new proof that Deligne's categories are generically semisimple. We also apply this method to Etingof's extrapolations of dAHA, and prove that when $n$ is transcendental, \"finite-dimensional\" simple objects are quotients of certain standard induced objects, extrapolating Zelevinsky's classification of simple dAHA-modules for $n\\in \\mathbb{N}$. Finally, we obtain similar results for the extrapolations of categories associated to wreath products of the symmetric group with associative algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-06-07T21:39:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "representation theory",
      "categories",
      "complex rank",
      "defined generically semisimple families",
      "degenerate affine hecke algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.1381M"
    }
  }
}