{
  "abstract": "Let $H_n(q)$ be the generic Iwahori--Hecke algebra associated to the symmetric group $\\mathfrak{S}_n$ of degree $n$ and let $\\mathscr{Y}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\\leq i\\leq \\binom{n}{2}$. We show that $\\mathscr{Y}_{n,1}$ and $\\mathscr{Y}_{n,2}$ are noncommutative whenever $n\\geq 4$, which disproves Doikou's Conjecture on the commutativity of the shuffle elements of $H_n(q)$ in (Nuclear Physics B 1029 (2026): 117532). We also show that the matrix of the operator of the left multiplication by $\\mathscr{Y}_{n,i}$ in $H_n(q)$ is $q$-symmetric for all $i$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11345",
  "author_total": 1,
  "authors": [
    {
      "name": "Deke Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1b4wybrtdb4jv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11345",
  "fulltext_url": null,
  "id": "1b4wybrtdb4jv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11345",
  "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.RT"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7212247153",
  "pdf_url": "https://arxiv.org/pdf/2609.11345",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1b4wybrtdb4jv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.11345",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11345"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Remarks on the shuffle elements of Iwahori--Hecke algebras",
  "title_text": "Remarks on the shuffle elements of Iwahori--Hecke algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/1b4wybrtdb4jv",
  "venue": "arXiv",
  "versions": []
}