{
  "id": "1908.00196",
  "version": "v1",
  "published": "2019-08-01T03:42:50.000Z",
  "updated": "2019-08-01T03:42:50.000Z",
  "title": "Alternating super-polynomials and super-coinvariants of finite reflection groups",
  "authors": [
    "Joshua P Swanson"
  ],
  "comment": "18 pages, 1 table",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "Motivated by a recent conjecture of Zabrocki, Wallach described the alternants in the super-coinvariant algebra of the symmetric group in one set of commuting and one set of anti-commuting variables under the diagonal action. We give a type-independent generalization of Wallach's result to all real reflection groups $G$. As an intermediate step, we explicitly describe the alternating super-polynomials in $k[V] \\otimes \\Lambda(V)$ for all complex reflection groups, providing an analogue of a classic result of Solomon which describes the invariant super-polynomials in $k[V] \\otimes \\Lambda(V^*)$. Using our construction, we explicitly describe the alternating harmonics and coinvariants for all real reflection groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-01T03:42:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10"
    ],
    "keywords": [
      "finite reflection groups",
      "alternating super-polynomials",
      "real reflection groups",
      "complex reflection groups",
      "invariant super-polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}