{
  "id": "2203.15213",
  "version": "v1",
  "published": "2022-03-29T03:14:53.000Z",
  "updated": "2022-03-29T03:14:53.000Z",
  "title": "Fans and polytopes in tilting theory",
  "authors": [
    "Toshitaka Aoki",
    "Akihiro Higashitani",
    "Osamu Iyama",
    "Ryoichi Kase",
    "Yuya Mizuno"
  ],
  "comment": "70 pages",
  "categories": [
    "math.RT",
    "math.CO",
    "math.CT",
    "math.RA"
  ],
  "abstract": "For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $\\Delta(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $\\Delta(A)$. Using $g$-vectors of 2-term silting complexes, $\\Delta(A)$ gives a nonsingular fan $\\Sigma(A)$ in the real Grothendieck group $K_0(\\mathrm{proj } A)_\\mathbb{R}$ called the $g$-fan. For example, the fan of $g$-vectors of a cluster algebra is given by the $g$-fan of a Jacobian algebra of a non-degenerate quiver with potential. We give several properties of $\\Sigma(A)$ including idempotent reductions, sign-coherence, Jasso reductions and a connection with Newton polytopes of $A$-modules. Moreover, $\\Sigma(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\\mathrm{proj } A)_\\mathbb{R}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. We give an explicit classification of $g$-convex algebras of rank $2$. We classify algebras whose $g$-polytopes are smooth Fano. We classify classical and generalized preprojective algebras which are $g$-convex, and also describe their $g$-polytope as the dual polytopes of short root polytopes of type $A$ and $B$. We also classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as root polytopes of type $A$ and $C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-29T03:14:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G20"
    ],
    "keywords": [
      "tilting theory",
      "simplicial complex",
      "dual polytope",
      "short root polytopes",
      "silting complexes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}