{
  "abstract": "We determine the Kazhdan--Lusztig polynomial $P_{\\mathrm{id},w}(q)$ for permutations $w$ such that $P_{\\mathrm{id},w}(1)=3$ and the Schubert variety $X_w$ has irreducible singular locus. Let $h=h(w)$ be the minimum height $w(a)-w(d)$ of a $3412$-occurrence in $w$ at positions $a<b<c<d$, with $h=1$ when no such occurrence exists. We show that $P_{\\mathrm{id},w}(q)=1+q^h+q^{h+1}$ if exactly one of the patterns $3412$ and $4231$ occurs, and $P_{\\mathrm{id},w}(q)=1+q+q^h$ if both patterns occur. In the latter case, $h>1$. This proves the irreducible singular locus case of a 2009 conjecture of Billey and Weed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32832",
  "author_total": 3,
  "authors": [
    {
      "name": "Gaston Burrull"
    },
    {
      "name": "Zhitong Alex Li"
    },
    {
      "name": "Shixuan Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4q431cz50m4kc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32832",
  "fulltext_url": null,
  "id": "4q431cz50m4kc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32832",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223663555",
  "pdf_url": "https://arxiv.org/pdf/2610.32832",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4q431cz50m4kc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32832",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32832"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Kazhdan--Lusztig Polynomials for permutations with $P_{\\mathrm{id},w}(1)=3$ and Irreducible Singular Locus",
  "title_text": "Kazhdan--Lusztig Polynomials for permutations with P_id,w(1)=3 and Irreducible Singular Locus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/4q431cz50m4kc",
  "venue": "arXiv",
  "versions": []
}