{
  "id": "1602.08837",
  "version": "v1",
  "published": "2016-02-29T06:21:47.000Z",
  "updated": "2016-02-29T06:21:47.000Z",
  "title": "Elementary symmetric polynomials in Stanley--Reisner face ring",
  "authors": [
    "Zhi Lü",
    "Jun Ma",
    "Yi Sun"
  ],
  "comment": "19 pages, 3 pictures",
  "categories": [
    "math.AT",
    "math.AC",
    "math.CO"
  ],
  "abstract": "Let $P$ be a simple polytope of dimension $n$ with $m$ facets. In this paper we pay our attention on those elementary symmetric polynomials in the Stanley--Reisner face ring of $P$ and study how the decomposability of the $n$-th elementary symmetric polynomial influences on the combinatorics of $P$ and the topology and geometry of toric spaces over $P$. We give algebraic criterions of detecting the decomposability of $P$ and determining when $P$ is $n$-colorable in terms of the $n$-th elementary symmetric polynomial. In addition, we define the Stanley--Reisner {\\em exterior} face ring $\\mathcal{E}(K_P)$ of $P$, which is non-commutative in the case of ${\\Bbb Z}$ coefficients, where $K_P$ is the boundary complex of dual of $P$. Then we obtain a criterion for the (real) Buchstaber invariant of $P$ to be $m-n$ in terms of the $n$-th elementary symmetric polynomial in $\\mathcal{E}(K_P)$. Our results as above can directly associate with the topology and geometry of toric spaces over $P$. In particular, we show that the decomposability of the $n$-th elementary symmetric polynomial in $\\mathcal{E}(K_P)$ with ${\\Bbb Z}$ coefficients can detect the existence of the almost complex structures of quasitoric manifolds over $P$, and if the (real) Buchstaber invariant of $P$ is $m-n$, then there exists an essential relation between the $n$-th equivariant characteristic class of the (real) moment-angle manifold over $P$ in $\\mathcal{E}(K_P)$ and the characteristic functions of $P$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-29T06:21:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stanley-reisner face ring",
      "th elementary symmetric polynomial influences",
      "toric spaces",
      "buchstaber invariant",
      "th equivariant characteristic class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160208837L"
    }
  }
}