{
  "id": "math/0303107",
  "version": "v1",
  "published": "2003-03-09T18:52:48.000Z",
  "updated": "2003-03-09T18:52:48.000Z",
  "title": "Ad-nilpotent ideals of a Borel subalgebra: generators and duality",
  "authors": [
    "Dmitri I. Panyushev"
  ],
  "comment": "LaTeX2e, 23 pages",
  "journal": "J. Algebra 274 (2004), 822-846",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "It was shown by Cellini and Papi that an ad-nilpotent ideal determines certain element of the affine Weyl group, and that there is a bijection between the ad-nilpotent ideals and the integral points of a simplex with rational vertices. We give a description of the generators of ad-nilpotent ideals in terms of these elements, and show that an ideal has $k$ generators if and only it lies on the face of this simplex of codimension $k$. We also consider two combinatorial statistics on the set of ad-nilpotent ideals: the number of simple roots in the ideal and the number of generators. Considering the first statistic reveals some relations with the theory of clusters (Fomin-Zelevinsky). The distribution of the second statistic suggests that there should exist a natural involution (duality) on the set of ad-nilpotent ideals. Such an involution is constructed for the series A,B,C.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-03-09T18:52:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "borel subalgebra",
      "generators",
      "affine weyl group",
      "ad-nilpotent ideal determines",
      "rational vertices"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......3107P"
    }
  }
}