{
  "abstract": "Haiman conjectured that for every partition $λ\\vdash n$ and every $w\\in S_n$ the polynomial $c_{λ,w}(q)$, a monomial character of the Hecke algebra evaluated at the Kazhdan-Lusztig basis, has non-negative integer coefficients and is symmetric and unimodal, and he verified this by computer for $n\\le7$. We verify it for every $n\\le10$. A stability lemma reduces the statement to $n=10$, which we settle by an exact computation that stores only the 1,252,173,030 pairs $(x,w)$ in $S_{10}$ with $x$ maximal in its double coset for the descent sets of $w$, and takes the traces of the standard basis from the class polynomials of Geck and Pfeiffer. Every stored Kazhdan-Lusztig polynomial agrees with du Cloux's Coxeter3, and the coefficients pass several independent checks, including a comparison at all 16,796 codominant $w$ with chromatic quasisymmetric functions computed from colourings. Consequently, for $n\\le10$ every monomial immanant is a non-negative integer combination of Kazhdan-Lusztig immanants, Stembridge's monomial immanant conjecture holds, and a class-function immanant is non-negative on totally non-negative matrices if and only if it is non-negative at the block matrices $J_μ$. We use this to describe the order of normalised irreducible-character immanants on totally non-negative matrices for $n\\le10$, and to prove Pate's inequalities there. On the full span of permutation monomials we give a Newton-polytope criterion for the coefficient of $\\mathrm{Imm}_w$ to be non-negative throughout the totally non-negative cone, show that for $n=4,5$ this happens exactly when $X_w$ is smooth, and check by exact computation for $n\\le10$ the proof of this for all $n$ given in a companion paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41555",
  "author_total": 1,
  "authors": [
    {
      "name": "Keith McNulty"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d4kw1wfsy7cr7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41555",
  "fulltext_url": null,
  "id": "d4kw1wfsy7cr7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41555",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41555",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d4kw1wfsy7cr7.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41555",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41555"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Haiman's conjecture for n <= 10: Kazhdan-Lusztig positivity of monomial immanants",
  "title_text": "Haiman's conjecture for n <= 10: Kazhdan-Lusztig positivity of monomial immanants",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/d4kw1wfsy7cr7",
  "venue": "arXiv",
  "versions": []
}