{
  "id": "2405.02092",
  "version": "v1",
  "published": "2024-05-03T13:35:06.000Z",
  "updated": "2024-05-03T13:35:06.000Z",
  "title": "Geometric realizations of the $s$-weak order and its lattice quotients",
  "authors": [
    "Eva Philippe",
    "Vincent Pilaud"
  ],
  "comment": "49 pages, 33 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For an $n$-tuple $s$ of non-negative integers, the $s$-weak order is a lattice structure on $s$-trees, generalizing the weak order on permutations. We first describe the join irreducible elements, the canonical join representations, and the forcing order of the $s$-weak order in terms of combinatorial objects, generalizing the arcs, the non-crossing arc diagrams, and the subarc order for the weak order. We then extend the theory of shards and shard polytopes to construct geometric realizations of the $s$-weak order and all its lattice quotients as polyhedral complexes, generalizing the quotient fans and quotientopes of the weak order.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-03T13:35:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06B10",
      "52B11",
      "52B12"
    ],
    "keywords": [
      "weak order",
      "lattice quotients",
      "construct geometric realizations",
      "subarc order",
      "combinatorial objects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}