{
  "id": "1111.6712",
  "version": "v2",
  "published": "2011-11-29T07:13:08.000Z",
  "updated": "2013-04-06T17:08:42.000Z",
  "title": "Modules of differential operators of order 2 on Coxeter arrangements",
  "authors": [
    "Norihiro Nakashima"
  ],
  "comment": "The title has changed. The previous title is \"Cauchy-Sylvester's theorem on compound determinants and modules of differential operators on Coxeter arrangements.\"",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that the modules of differential operators of order 2 on the classical Coxeter arrangements are free by exhibiting bases. For this purpose, we use Cauchy-Sylvester's theorem on compound determinants and Saito-Holm's criterion. In the case type $A$, we apply Cauchy-Sylvester's theorem on compound determinants to Vandermond determinant. By using the Schur polynomials, we define operators which form a part of a basis of modules of differential operators on the classical Coxeter arrangements of type $A$. In the cases of type $B$ and type $D$, the proofs go similarly to the case of type $A$ with some adjustments of operators and determinants.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-06T17:08:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S22",
      "15A15"
    ],
    "keywords": [
      "differential operators",
      "classical coxeter arrangements",
      "compound determinants",
      "define operators",
      "schur polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.6712N"
    }
  }
}