{
  "id": "2111.13540",
  "version": "v2",
  "published": "2021-11-26T15:19:26.000Z",
  "updated": "2022-11-23T21:55:03.000Z",
  "title": "Complexity of the usual torus action on Kazhdan-Lusztig varieties",
  "authors": [
    "Maria Donten-Bury",
    "Laura Escobar",
    "Irem Portakal"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "abstract": "We investigate the class of Kazhdan-Lusztig varieties, and its subclass of matrix Schubert varieties, endowed with a naturally defined torus action. Writing a matrix Schubert variety $\\overline{X_w}$ as $\\overline{X_w}=Y_w\\times \\mathbb{C}^d$ (where $d$ is maximal possible), we show that $Y_w$ can be of complexity-$k$ exactly when $k\\neq 1$. Also, we give a combinatorial description of the extremal rays of the weight cone of a Kazhdan-Lusztig variety, which in particular turns out to be the edge cone of an acyclic directed graph. As a consequence we show that given permutations $v$ and $w$, the complexity of Kazhdan-Lusztig variety indexed by $(v,w)$ is the same as the complexity of the Richardson variety indexed by $(v,w)$. Finally, we use this description to compute the complexity of certain Kazhdan-Lusztig varieties.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-11-23T21:55:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "14M25",
      "52B20",
      "05E10",
      "05C20"
    ],
    "keywords": [
      "kazhdan-lusztig variety",
      "usual torus action",
      "complexity",
      "matrix schubert variety",
      "acyclic directed graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}