{
  "id": "2107.07428",
  "version": "v1",
  "published": "2021-07-15T16:17:51.000Z",
  "updated": "2021-07-15T16:17:51.000Z",
  "title": "Subrepresentations in the homology of finite covers of graphs",
  "authors": [
    "Xenia Flamm"
  ],
  "comment": "14 pages. Comments welcome!",
  "categories": [
    "math.GT",
    "math.RT"
  ],
  "abstract": "Let $p \\colon Y \\to X$ be a finite, regular cover of finite graphs with associated deck group $G$, and consider the first homology $H_1(Y;\\mathbb{C})$ of the cover as a $G$-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group $G$ on the one hand, and topological properties of homology classes in $H_1(Y;\\mathbb{C})$ on the other hand. We do so by studying certain subrepresentations in the $G$-representation $H_1(Y;\\mathbb{C})$. The homology class of a lift of a primitive element in $\\pi_1(X)$ spans an induced subrepresentation in $H_1(Y;\\mathbb{C})$, and we show that this property is never sufficient to characterize such homology classes if $G$ is Abelian. We study $H_1^{\\textrm{comm}}(Y;\\mathbb{C}) \\leq H_1(Y;\\mathbb{C})$ -- the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in $\\pi_1(X)$. Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with $H_1^{\\textrm{comm}}(Y;\\mathbb{C}) \\neq \\ker(p_*)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-15T16:17:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M10",
      "57M60",
      "20C15"
    ],
    "keywords": [
      "homology class",
      "finite covers",
      "subrepresentation",
      "primitive element",
      "construct examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}