{
  "id": "math/0310121",
  "version": "v1",
  "published": "2003-10-08T18:26:07.000Z",
  "updated": "2003-10-08T18:26:07.000Z",
  "title": "The cd-index of Bruhat intervals",
  "authors": [
    "Nathan Reading"
  ],
  "comment": "25 pages, 6 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index. This leads to a new proof that Bruhat intervals are PL spheres as well a recursive formula for the cd-index of a Bruhat interval. This recursive formula is used to prove that the cd-indices of Bruhat intervals span the space of cd-polynomials. The structural recursion leads to a conjecture that Bruhat spheres are \"smaller\" than polytopes. More precisely, we conjecture that if one fixes the lengths of x and y, then the cd-index of a certain dual stacked polytope is a coefficientwise upper bound on the cd-indices of Bruhat intervals [x,y]. We show that this upper bound would be tight by constructing Bruhat intervals which are the face lattices of these dual stacked polytopes. As a weakening of a special case of the conjecture, we show that the flag h-vectors of lower Bruhat intervals are bounded above by the flag h-vectors of Boolean algebras (i.e. simplices).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-10-08T18:26:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "06A07"
    ],
    "keywords": [
      "dual stacked polytope",
      "bruhat order",
      "flag h-vectors",
      "structural recursion",
      "upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....10121R"
    }
  }
}