{
  "id": "2010.10582",
  "version": "v1",
  "published": "2020-10-20T19:40:16.000Z",
  "updated": "2020-10-20T19:40:16.000Z",
  "title": "Stability of stretched root systems, root posets, and shards",
  "authors": [
    "Will Dana"
  ],
  "comment": "23 pages, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Inspired by the infinite families of finite and affine root systems, we consider a \"stretching\" operation on general crystallographic root systems which, on the level of Coxeter diagrams, replaces a vertex with a path of unlabeled edges. We embed a root system into its stretched versions using a similar operation on individual roots. For a fixed root, we study the growth of two associated structures as we lengthen the stretched path: the downset in the root poset (in the sense of Bj\\\"orner and Brenti [3]) and the arrangement of shards, introduced by Nathan Reading. We show that both eventually admit a uniform description, and deduce enumerative consequences: the size of the downset is eventually a polynomial, and the number of shards grows exponentially.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-20T19:40:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B22",
      "20F55",
      "52C35"
    ],
    "keywords": [
      "stretched root systems",
      "root poset",
      "general crystallographic root systems",
      "affine root systems",
      "infinite families"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}