{
  "id": "1002.0810",
  "version": "v1",
  "published": "2010-02-03T17:58:33.000Z",
  "updated": "2010-02-03T17:58:33.000Z",
  "title": "Ring of Polytopes, Quasi-symmetric functions and Fibonacci numbers",
  "authors": [
    "Victor M. Buchstaber",
    "Nickolai Erokhovets"
  ],
  "comment": "42 pages",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "abstract": "In this paper we study the ring $\\mathcal{P}$ of combinatorial convex polytopes. We introduce the algebra of operators $\\mathcal{D}$ generated by the operators $d_k$ that send an $n$-dimensional polytope $P^n$ to the sum of all its $(n-k)$-dimensional faces. It turns out that $\\mathcal{D}$ is isomorphic to the universal Leibnitz-Hopf algebra with the antipode $\\chi(d_k)=(-1)^kd_k$. Using the operators $d_k$ we build the generalized $f$-polynomial, which is a ring homomorphism from $\\mathcal{P}$ to the ring $\\Qsym[t_1,t_2,...][\\alpha]$ of quasi-symmetric functions with coefficients in $\\mathbb Z[\\alpha]$. The images of two polytopes coincide if and only if their flag $f$-vectors are equal. We describe the image of this homomorphism over the integers and prove that over the rationals it is a free polynomial algebra with dimension of the $n$-th graded component equal to the $n$-th Fibonacci number. This gives a representation of the Fibonacci series as an infinite product. The homomorphism is an isomorphism on the graded group $BB$ generated by the polytopes introduced by Bayer and Billera to find the linear span of flag $f$-vectors of convex polytopes. This gives the group $BB$ a structure of the ring isomorphic to $f(\\mathcal{P})$. We show that the ring of polytopes has a natural Hopf comodule structure over the Rota-Hopf algebra of posets. As a corollary we build a ring homomorphism $l_{\\alpha}\\colon\\mathcal{P}\\to\\mathcal{R}[\\alpha]$ such that $F(l_{\\alpha}(P))=f(P)^*$, where $F$ is the Ehrenborg quasi-symmetric function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-03T17:58:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "52B05",
      "16T05",
      "57T25"
    ],
    "keywords": [
      "natural hopf comodule structure",
      "combinatorial convex polytopes",
      "universal leibnitz-hopf algebra",
      "th fibonacci number",
      "ring homomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.0810B"
    }
  }
}