{
  "abstract": "Association schemes induced on the shells of the Grassmann scheme $Gr_q(N,a)$ form an important family in the study of multivariate $P$- and $Q$-polynomial structures. For $a<N/2$, the scheme induced on the furthest shell $X_a$ is the known bivariate $P$- and $Q$-polynomial scheme arising from attenuated spaces. In this paper, we prove that, for every $1\\leq i\\leq a-1$, the association scheme induced on the interior shell $X_i$ is both trivariate $P$-polynomial and trivariate $Q$-polynomial. The two structures share a common index set consisting of the lattice points of a bounded non-simplicial polyhedron. We establish the $P$-polynomial property through combinatorial interpretations of the intersection numbers. Using Dunkl's coordinate description of $X_i$ and the addition theorem, together with recurrence relations for the $q$-Hahn and $q$-Krawtchouk polynomials, we obtain the required vanishing and nonvanishing conditions for the Krein parameters, and hence the $Q$-polynomial structure. Motivated by this index set, we extend the notion of an $A_M$-Leonard pair from simplicial to polyhedral domains and show that every interior shell carries a polyhedral $A_3$-Leonard pair.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19974",
  "author_total": 1,
  "authors": [
    {
      "name": "Yan Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkqp4jv7k2dqb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19974",
  "fulltext_url": null,
  "id": "dkqp4jv7k2dqb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19974",
  "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"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7222194573",
  "pdf_url": "https://arxiv.org/pdf/2610.19974",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkqp4jv7k2dqb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19974",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19974"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Trivariate $P$- and $Q$-polynomial structures for interior shells of the Grassmann scheme",
  "title_text": "Trivariate P- and Q-polynomial structures for interior shells of the Grassmann scheme",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/dkqp4jv7k2dqb",
  "venue": "arXiv",
  "versions": []
}