{
  "id": "0709.4421",
  "version": "v2",
  "published": "2007-09-27T15:30:04.000Z",
  "updated": "2008-04-17T17:57:00.000Z",
  "title": "Isometry classes of generalized associahedra",
  "authors": [
    "Nantel Bergeron",
    "Christophe Hohlweg",
    "Carsten Lange",
    "Hugh Thomas"
  ],
  "comment": "12 pages, 4 figures, pdflatex: v2: correction of typos",
  "journal": "S\\'em. Lothar. Combin. 61A (2009), Art. B61Aa, 13 pp.",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\\mathbb R$-Euclidean space with simple roots $\\Delta=\\{\\a_s | s\\in S\\}$ of the same length and fundamental weights $\\Delta^*=\\{v_s | s\\in S\\}$. We set $M(e)=\\sum_{s\\in S}\\kappa_s v_s$, $\\kappa_s>0$, and for $w\\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\\{M(w) | w\\in W\\}$. Given a Coxeter element $c\\in W$, we have defined in a previous work a generalized associahedron $Asso_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Asso_c(W)$ is obtained from $Perm(W)$ by removing some halfspaces according to a rule prescribed by $c$. In this work, we classify the isometry classes of these realizations. More precisely, for $(W,S)$ an irreducible finite Coxeter system and $c,c'$ two Coxeter elements in $W$, we have that $Asso_{c}(W)$ and $Asso_{c'}(W)$ are isometric if and only if $\\mu(c') = c$ or $\\mu(c')=w_0c^{-1}w_0$ for $\\mu$ an automorphism of the Coxeter graph of $W$ such that $\\kappa_s=\\kappa_{\\mu(s)}$ for all $s\\in S$. As a byproduct, we classify the isometric Cambrian fans of $W$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-04-17T17:57:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F55",
      "06B99",
      "52B11",
      "05E99"
    ],
    "keywords": [
      "isometry classes",
      "generalized associahedron",
      "coxeter element",
      "irreducible finite coxeter system",
      "isometric cambrian fans"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "PDFLaTeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.4421B"
    }
  }
}