{
  "id": "0909.1612",
  "version": "v1",
  "published": "2009-09-09T02:25:43.000Z",
  "updated": "2009-09-09T02:25:43.000Z",
  "title": "$q,t$-Catalan numbers and generators for the radical ideal defining the diagonal locus of $(\\C^2)^n$",
  "authors": [
    "Kyungyong Lee",
    "Li Li"
  ],
  "comment": "29 pages",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Let $I$ be the ideal generated by alternating polynomials in two sets of $n$ variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the graded vector space $M(=\\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by a minimal set of generators for $I$. In this paper we give simple upper bounds on $\\text{dim}M_{d_1, d_2}$ in terms of partition numbers, and find all bi-degrees $(d_1,d_2)$ such that $\\dim M_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $M_{d_1, d_2}$. The main idea is to define and study a nontrivial linear map from $M$ to a polynomial ring $\\C[\\rho_1, \\rho_2,...]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-09-09T02:25:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15",
      "05E40"
    ],
    "keywords": [
      "catalan number",
      "radical ideal defining",
      "diagonal locus",
      "generators",
      "nontrivial linear map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.1612L"
    }
  }
}