{
  "id": "2505.06147",
  "version": "v1",
  "published": "2025-05-09T15:58:08.000Z",
  "updated": "2025-05-09T15:58:08.000Z",
  "title": "A categorification of combinatorial Auslander-Reiten quivers",
  "authors": [
    "Ricardo Canesin"
  ],
  "comment": "49 pages, 2 figures",
  "categories": [
    "math.RT"
  ],
  "abstract": "We provide a categorification of Oh and Suh's combinatorial Auslander-Reiten quivers in the simply laced case. We work within the perfectly valued derived category $\\mathrm{pvd}(\\Pi_Q)$ of the 2-dimensional Ginzburg dg algebra of a Dynkin quiver $Q$. For any commutation class $[i]$ of reduced words in the corresponding Weyl group, we define a subcategory $C([i])$ of $\\mathrm{pvd}(\\Pi_Q)$ whose objects are obtained by applying a sequence of spherical twist functors to the simple objects. We describe the Hom-order for $C([i])$ in terms of $[i]$, generalizing a result of B\\'edard. Furthermore, when $[i]$ is a commutation class for the longest element, we construct a category $D([i])$ generalizing the bounded derived category of $Q$. It is realized as a certain subquotient of $\\mathrm{pvd}(\\Pi_Q)$. We demonstrate the existence of particular distinguished triangles in $\\mathrm{pvd}(\\Pi_Q)$ with corners in $D([i])$, which allows us to extend the classical mesh-additivity to arbitrary commutation classes. Additionally, we define an analog of the Euler form and prove that its symmetrization yields the corresponding Cartan-Killing form. For commutation classes $[i]$ arising from Q-data, a generalization of Dynkin quivers with a height function introduced by Fujita and Oh, we establish the existence of a partially Serre functor on $D([i])$. Lastly, we apply our results to reinterpret a formula by Fujita and Oh for the inverse of the quantum Cartan matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-09T15:58:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "18G80",
      "20F55"
    ],
    "keywords": [
      "categorification",
      "dynkin quiver",
      "suhs combinatorial auslander-reiten quivers",
      "derived category",
      "quantum cartan matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}