{
  "id": "2406.19962",
  "version": "v1",
  "published": "2024-06-28T14:48:51.000Z",
  "updated": "2024-06-28T14:48:51.000Z",
  "title": "Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids",
  "authors": [
    "Luis Ferroni",
    "Jacob P. Matherne",
    "Lorenzo Vecchi"
  ],
  "comment": "14 pages. A part of this article was previously included in arXiv:2212.03190",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "We study equivariant Kazhdan--Lusztig (KL) and $Z$-polynomials of matroids. We formulate an equivariant generalization of a result by Braden and Vysogorets that relates the equivariant KL and $Z$-polynomials of a matroid with those of a single-element deletion. We also discuss the failure of equivariant $\\gamma$-positivity for the $Z$-polynomial. As an application of our main result, we obtain a formula for the equivariant KL polynomial of the graphic matroid gotten by gluing two cycles. Furthermore, we compute the equivariant KL polynomials of all matroids of corank~$2$ via valuations. This provides an application of the machinery of Elias, Miyata, Proudfoot, and Vecchi to corank $2$ matroids, and it extends results of Ferroni and Schr\\\"oter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-28T14:48:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equivariant kazhdan-lusztig polynomials",
      "deletion formulas",
      "equivariant kl polynomial",
      "study equivariant kazhdan-lusztig",
      "graphic matroid gotten"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}