{
  "id": "2303.16569",
  "version": "v1",
  "published": "2023-03-29T10:06:45.000Z",
  "updated": "2023-03-29T10:06:45.000Z",
  "title": "Shi arrangements and low elements in Coxeter groups",
  "authors": [
    "Matthew Dyer",
    "Christophe Hohlweg",
    "Susanna Fishel",
    "Alice Mark"
  ],
  "comment": "41 pages, 7 figures",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Given an arbitrary Coxeter system $(W,S)$ and a nonnegative integer $m$, the $m$-Shi arrangement of $(W,S)$ is a subarrangement of the Coxeter hyperplane arrangement of $(W,S)$. The classical Shi arrangement ($m=0$) was introduced in the case of affine Weyl groups by Shi to study Kazhdan-Lusztig cells for $W$. As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in $W$ and that the union of their inverses form a convex subset of the Coxeter complex. The set of $m$-low elements in $W$ were introduced to study the word problem of the corresponding Artin-Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in $W$. In this article, we generalize and extend Shi's results to any Coxeter system for any $m$: (1) the set of minimal length elements of the regions in a $m$-Shi arrangement is precisely the set of $m$-low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the ($0$-)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-29T10:06:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "05E16",
      "17B22",
      "06A07",
      "06A11"
    ],
    "keywords": [
      "coxeter groups",
      "convex subset",
      "third author",
      "coxeter complex",
      "coxeter hyperplane arrangement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}