{
  "id": "1404.4260",
  "version": "v3",
  "published": "2014-04-16T14:19:05.000Z",
  "updated": "2015-10-29T07:32:36.000Z",
  "title": "C-vectors via $τ$-tilting theory",
  "authors": [
    "Changjian Fu"
  ],
  "comment": "27pages, references added, corrected some typos",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "Inspired by the tropical duality in cluster algebras, we introduce c-vectors for finite-dimensional algebras via $\\tau$-tilting theory. Let $A$ be a finite-dimensional algebra over a field $k$. Each c-vector of $A$ can be realized as the (negative) dimension vector of certain indecomposable $A$-module and hence we establish the sign-coherence property of this kind of $c$-vectors. We then study the positive c-vectors for certain classes of finite-dimensional algebras. More precisely, we establish the equalities between the set of positive c-vectors and the set of dimension vectors of exceptional modules for quasitilted algebras and representation-directed algebras respectively. This generalizes the equalitites of c-vectors for acyclic cluster algebras obtained by Ch\\'{a}vez. To this end, a short proof for the sign-coherence of c-vectors for skew-symmetric cluster algebras has been given in the appendix.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-24T04:21:06.000Z",
      "comment": "24pages, references added, corrected some typos",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-10-29T07:32:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tilting theory",
      "finite-dimensional algebra",
      "dimension vector",
      "positive c-vectors",
      "skew-symmetric cluster algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.4260F"
    }
  }
}