{
  "id": "2104.14531",
  "version": "v1",
  "published": "2021-04-29T17:47:37.000Z",
  "updated": "2021-04-29T17:47:37.000Z",
  "title": "Matroid relaxations and Kazhdan-Lusztig non-degeneracy",
  "authors": [
    "Luis Ferroni",
    "Lorenzo Vecchi"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we address a variant of the Kazhdan-Lusztig non-degeneracy conjecture posed by Gedeon, Proudfoot and Young. We prove that if $M$ has a free basis (something that conjecturally asymptotically all matroids are expected to possess), then $M$ is non-degenerate. To this end, we study the behavior of Kazhdan-Lusztig polynomials of matroids with respect to the operation of circuit-hyperplane relaxation. This yields a family of polynomials that relate the Kazhdan-Lusztig, the inverse Kazhdan-Lusztig and the $Z$-polynomial of a matroid with those of its relaxations and do not depend on the matroid. As an application of our results, we deduce that uniform matroids maximize coefficient-wise the Kazhdan-Lusztig polynomials, inverse Kazhdan-Lusztig polynomials and the $Z$-polynomials, when restricted to sparse paving matroids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-29T17:47:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "matroid relaxations",
      "kazhdan-lusztig non-degeneracy conjecture",
      "inverse kazhdan-lusztig polynomials",
      "circuit-hyperplane relaxation",
      "uniform matroids"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}