{
  "id": "2410.12231",
  "version": "v1",
  "published": "2024-10-16T04:46:09.000Z",
  "updated": "2024-10-16T04:46:09.000Z",
  "title": "A geometric realization of the chromatic symmetric function of a unit interval graph",
  "authors": [
    "Syu Kato"
  ],
  "comment": "16pp",
  "categories": [
    "math.CO",
    "math.AG",
    "math.RT"
  ],
  "abstract": "Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math.$ {\\bf 295}$ (2016), ${\\bf 329}$ (2018), arXiv:1601.05498] realized the chromatic (quasi-)symmetric function of a unit interval graph in terms of Hessenberg varieties. Here we exhibit another realization of these chromatic (quasi-)symmetric functions in terms of the Betti cohomology of the variety $\\mathscr X_\\Psi$ defined in [arXiv:2301.00862]. This yields a new inductive combinatorial expression of these chromatic symmetric functions. Based on these, we propose a geometric refinement of the Stanley-Stembridge conjecture, whose validity would imply the Sharesian-Wachs conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-16T04:46:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "chromatic symmetric function",
      "unit interval graph",
      "geometric realization",
      "stanley-stembridge conjecture",
      "geometric refinement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}