{
  "abstract": "Given a finite Coxeter group $W$ and an element $q$ of a commutative ring $\\mathbf{k}$, we define the element $\\mathbf{L}_{q} :=\\sum_{w\\in W}q^{\\ell\\left( w\\right) }w$ of its group algebra $\\mathbf{k}\\left[ W\\right] $. We say that an element $w\\in W$ is $W$\\emph{-length-symmetric} if its coefficient in the commutator $\\left[ \\mathbf{L}_{a},\\mathbf{L}_{b}\\right] $ is $0$ for all $\\mathbf{k}$ and all $a,b\\in\\mathbf{k}$. We prove several sufficient criteria for elements to be $W$-length-symmetric. In particular, we show that any separable permutation (i.e., permutation avoiding the patterns $2413$ and $3142$) in the symmetric group $S_{n}$ is $S_{n}$-length-symmetric. More generally, we construct a notion of separable elements of a finite Coxeter group $W$ (which, for finite Weyl groups, agrees with the eponymous notion introduced by Gaetz and Gao), and show that all such elements are $W$-length-symmetric. This is only a sufficient condition; other $W$-length-symmetric elements include (twisted) involutions and elements of dihedral parabolic subgroups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07365",
  "author_total": 2,
  "authors": [
    {
      "name": "Ben Adenbaum"
    },
    {
      "name": "Darij Grinberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckdbq9tyzsjvh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07365",
  "fulltext_url": null,
  "id": "ckdbq9tyzsjvh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07365",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07365",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckdbq9tyzsjvh.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07365",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07365"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The $q$-Poincaré sum and separability in a Coxeter group",
  "title_text": "The q-Poincaré sum and separability in a Coxeter group",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/ckdbq9tyzsjvh",
  "venue": "arXiv",
  "versions": []
}