{
  "id": "1306.1593",
  "version": "v2",
  "published": "2013-06-07T02:41:12.000Z",
  "updated": "2018-01-20T07:52:28.000Z",
  "title": "The root posets and their rich antichains",
  "authors": [
    "Claus Michael Ringel"
  ],
  "comment": "This is a completely revised version, now with reference to the exponents. The (n-1)-antichains are now called rich antichains and there is an outline in which way the root poset can be recovered from the set of rich antichains",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "Let $\\Delta$ be a (connected) Dynkin diagram of rank $n\\ge 2$ and $\\Phi_+ = \\Phi_+(\\Delta)$ the corresponding root poset (it consists of all positive roots with respect to a fixed root basis). The width of $\\Phi_+$ is $n$. We will show that $\\Phi_+$ is \"conical\": it is the disjoint union of $n$ solid chains. The rich antichains in $\\Phi_+$ are the antichains of cardinality $n-1$. It is well known that the number of rich antichains is equal to the cardinality of $\\Phi_+$. The set $\\mathcal R(\\Delta)$ of rich antichains in $\\Phi_+$ can itself be considered as a poset which is quite similar, but not always isomorphic, to $\\Phi_+$. We will show that there always exists a unique rich antichain $A$ such that any rich antichain is contained in the ideal generated by $A$. For $\\Delta\\neq \\Bbb E_6$ all roots in $A$ have the same length, namely $e_2$, where $e_1 \\le e_2 \\le \\dots \\le e_n$ are the exponents of $\\Delta.$ For $\\Delta = \\Bbb E_6$, the antichain $A$ consists of four roots of length $e_2 = 4$ and one root of length $5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-07T02:41:12.000Z",
      "title": "The (n-1)-antichains in a root poset of width n",
      "abstract": "Let $\\Delta$ be a Dynkin diagram of rank n > 1 and $\\Phi_+$ the corresponding root poset (it consists of all positive roots with respect to some root basis). Antichains in $\\Phi_+$ of cardinality t will be called t-antichains. There always exists a unique maximal (n-1)-antichain A. We will show that if $\\Delta$ is not of type $E_6,$ then all roots in A have the same length with respect to the root basis, whereas for Dynkin type $E_6$, this antichain consists of roots of length 4 and 5. It follows from these considerations that in general one cannot recover the poset structure of $\\Phi_+$ by looking at the set of (n-1)-antichains in $\\Phi_+$.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-01-20T07:52:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "root basis",
      "corresponding root poset",
      "poset structure",
      "dynkin diagram",
      "unique maximal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.1593R"
    }
  }
}