{
  "id": "2205.09854",
  "version": "v1",
  "published": "2022-05-19T20:59:47.000Z",
  "updated": "2022-05-19T20:59:47.000Z",
  "title": "Symmetric products of dg categories and semi-orthogonal decompositions",
  "authors": [
    "Naoki Koseki"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AG",
    "math.CT",
    "math.RT"
  ],
  "abstract": "In this article, we investigate semi-orthogonal decompositions of the symmetric products of dg-enhanced triangulated categories. Given a semi-orthogonal decomposition $\\mathcal{D}=\\langle \\mathcal{A}, \\mathcal{B} \\rangle$, we construct semi-orthogonal decompositions of the symmetric products of $\\mathcal{D}$ in terms of that of $\\mathcal{A}$ and $\\mathcal{B}$. This was originally stated by Galkin--Shinder, and answers the question raised by Ganter--Kapranov. We give two applications of our main result. Firstly, combining the above result with the derived McKay correspondence, we obtain various interesting semi-orthogonal decompositions of the derived categories of the Hilbert schemes of points on surfaces. Secondly, we prove the compatibility of our semi-orthogonal decompositions with categorical Heisenberg actions on the symmetric products of dg-categories due to Gyenge--Koppensteiner--Logvinenko. Using this compatibility, we prove the blow-up formula for the Heisenberg representations on the Grothendieck groups of the Hilbert schemes of points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-19T20:59:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric products",
      "dg categories",
      "hilbert schemes",
      "construct semi-orthogonal decompositions",
      "derived mckay correspondence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}