{
  "id": "1103.1085",
  "version": "v2",
  "published": "2011-03-05T21:45:54.000Z",
  "updated": "2012-02-18T22:26:04.000Z",
  "title": "Kleiner's theorem for unitary representations of posets",
  "authors": [
    "Yurii Samoilenko",
    "Kostyantyn Yusenko"
  ],
  "comment": "12 pages, paper reorganized and rewritten. some statements were added",
  "categories": [
    "math.RT",
    "math.FA"
  ],
  "abstract": "A subspace representation of a poset $\\mathcal S=\\{s_1,...,s_t\\}$ is given by a system $(V;V_1,...,V_t)$ consisting of a vector space $V$ and its subspaces $V_i$ such that $V_i\\subseteq V_j$ if $s_i \\prec s_j$. For each real-valued vector $\\chi=(\\chi_1,...,\\chi_t)$ with positive components, we define a unitary $\\chi$-representation of $\\mathcal S$ as a system $(U;U_1,...,U_t)$ that consists of a unitary space $U$ and its subspaces $U_i$ such that $U_i\\subseteq U_j$ if $s_i\\prec s_j$ and satisfies $\\chi_1 P_1+...+\\chi_t P_t= \\mathbb 1$, in which $P_i$ is the orthogonal projection onto $U_i$. We prove that $\\mathcal S$ has a finite number of unitarily nonequivalent indecomposable $\\chi$-representations for each weight $\\chi$ if and only if $\\mathcal S$ has a finite number of nonequivalent indecomposable subspace representations; that is, if and only if $\\mathcal S$ contains any of Kleiner's critical posets.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-02-18T22:26:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A63",
      "15A21",
      "16G20"
    ],
    "keywords": [
      "unitary representations",
      "kleiners theorem",
      "finite number",
      "nonequivalent indecomposable subspace representations",
      "vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.1085S"
    }
  }
}