{
  "abstract": "We determine Lusztig's asymptotic algebra for the second lowest two-sided cell~$Δ_1$ in the equal parameter extended affine Weyl group $\\tilde{\\mathsf{B}}_n$ for all $n\\geq 2$, realising it as a full matrix algebra over the representation ring of $\\mathsf{Sp}_{2n-2}(\\CC)\\times C_2$ with $C_2$ the group of order~$2$. Our result is an explicit isomorphism of based rings, determining the image of each basis element $\\mathsf{t}_w$ of the asymptotic algebra. As a consequence we verify that Lusztig's original conjecture from 1989 on the structure of the asymptotic algebra holds for this cell. A key ingredient in our proof is a refinement of Shi's description of~$Δ_1$, giving an explicit parametrisation of the elements of this cell (a \\textit{cell factorisation}). This cell factorisation simultaneously describes all intersections of left and right cells, with uniqueness and length-additivity of expressions. Applications include a classification of the involutions in $Δ_1$, and a calculation of the length generating functions for $Δ_1$ and the right cells in~$Δ_1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25771",
  "author_total": 1,
  "authors": [
    {
      "name": "James Parkinson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dxhk2kh1bcr4m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25771",
  "fulltext_url": null,
  "id": "dxhk2kh1bcr4m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25771",
  "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": "W7223048152",
  "pdf_url": "https://arxiv.org/pdf/2610.25771",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dxhk2kh1bcr4m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25771",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25771"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Lusztig's asymptotic algebra and a cell factorisation for the second lowest two-sided cell in type $\\tilde{B}_n$",
  "title_text": "Lusztig's asymptotic algebra and a cell factorisation for the second lowest two-sided cell in type B̃_n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/dxhk2kh1bcr4m",
  "venue": "arXiv",
  "versions": []
}