{
  "id": "1209.1659",
  "version": "v2",
  "published": "2012-09-07T22:02:22.000Z",
  "updated": "2013-11-07T14:10:20.000Z",
  "title": "Commuting varieties of $r$-tuples over Lie algebras",
  "authors": [
    "Nham V. Ngo"
  ],
  "comment": "To appear in Journal of Pure and Applied Algebra",
  "doi": "10.1016/j.jpaa.2013.11.024",
  "categories": [
    "math.RT",
    "math.AC"
  ],
  "abstract": "Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ of characteristic $p$ and let $\\g$ be the Lie algebra of $G$. It is well known that for $p$ large enough the spectrum of the cohomology ring for the $r$-th Frobenius kernel of $G$ is homeomorphic to the commuting variety of $r$-tuples of elements in the nilpotent cone of $\\g$ [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, \\textbf{10} (1997), 693--728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties $C_r(\\mathfrak{gl}_2), C_r(\\fraksl_2)$ and $C_r(\\N)$ where $\\N$ is the nilpotent cone of $\\fraksl_2$. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of $r$-tuples. Furthermore, in the case when $\\g=\\fraksl_2$, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of $\\fraksl_3$, we are able to verify the aforementioned properties for $C_r(\\fraku)$. Finally, applying our calculations on the commuting variety $C_r(\\overline{\\calO_{\\sub}})$ where $\\overline{\\calO_{\\sub}}$ is the closure of the subregular orbit in $\\fraksl_3$, we prove that the nilpotent commuting variety $C_r(\\N)$ has singularities of codimension $\\ge 2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-07T14:10:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20Gxx",
      "13C14"
    ],
    "keywords": [
      "lie algebra",
      "nilpotent cone",
      "th frobenius kernel",
      "nilpotent commuting variety",
      "simple algebraic group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.1659N"
    }
  }
}