{
  "id": "0905.0377",
  "version": "v2",
  "published": "2009-05-04T12:53:11.000Z",
  "updated": "2010-03-30T15:25:06.000Z",
  "title": "Basis of Diagonally Alternating Harmonic Polynomials for low degree",
  "authors": [
    "Nantel Bergeron",
    "Zhi Chen"
  ],
  "comment": "To appear in JCT-A; 21 pages, one PDF figure",
  "journal": "JCT-A 118 (2011) 37--57",
  "doi": "10.1016/j.jcta.2010.04.002",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Given a list of $n$ cells $L=[(p_1,q_1),...,(p_n, q_n)]$ where $p_i, q_i\\in \\textbf{Z}_{\\ge 0}$, we let $\\Delta_L=\\det |{(p_j!)^{-1}(q_j!)^{-1} x^{p_j}_iy^{q_j}_i} |$. The space of diagonally alternating polynomials is spanned by $\\{\\Delta_L\\}$ where $L$ varies among all lists with $n$ cells. For $a>0$, the operators $E_a=\\sum_{i=1}^{n} y_i\\partial_{x_i}^a$ act on diagonally alternating polynomials and Haiman has shown that the space $A_n$ of diagonally alternating harmonic polynomials is spanned by $\\{E_\\lambda\\Delta_n\\}$. For $t=(t_m,...,t_1)\\in \\textbf{Z}_{> 0}^m$ with $t_m>...>t_1>0$, we consider here the operator $F_t=\\det\\big\\|E_{t_{m-j+1}+(j-i)}\\big\\|$. Our first result is to show that $F_t\\Delta_L$ is a linear combination of $\\Delta_{L'}$ where $L'$ is obtained by {\\sl moving} $\\ell(t)=m$ distinct cells from $L$ in some determined fashion. This allows us to control the leading term of some elements of the form $F_{t_{(1)}}... F_{t_{(r)}}\\Delta_n$. We use this to describe explicit bases of some of the bihomogeneous components of $A_n=\\bigoplus A_n^{k,l}$ where $A_n^{k,l}=\\hbox{Span}\\{E_\\lambda\\Delta_n :\\ell(\\lambda)=l, |\\lambda|=k\\}$. More precisely we give an explicit basis of $A_n^{k,l}$ whenever $k<n$. To this end, we introduce a new variation of Schensted insertion on a special class of tableaux. This produces a bijection between partitions and this new class of tableaux. The combinatorics of those tableaux $T$ allows us to know exactly the leading term of $F_T\\Delta_n$ where $F_T$ is the operator corresponding to the columns of $T$ and whenever $n$ is bigger than the weight of $T$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-03-30T15:25:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "14M99"
    ],
    "keywords": [
      "diagonally alternating harmonic polynomials",
      "low degree",
      "diagonally alternating polynomials",
      "leading term",
      "linear combination"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.0377B"
    }
  }
}