{
  "abstract": "Mitra, Offen, and Sayag introduced distinguished multisegments and multisegments of Speh type, and proposed that every distinguished multisegment is of Speh type, together with a related duality hypothesis (see A. Mitra, O. Offen, and E. Sayag, Klyachko Models for Ladder Representations, Documenta Math. 22 (2017), 611-657). They proved these statements for sets of segments and when at most two segments share an endpoint. We develop a combinatorial theory of relevant decompositions and prove that relevance is preserved under the involution $Δ\\mapstoΔ^\\vee$ together with reversal of standard order. Hence, $\\mathfrak m$ is distinguished if and only if $\\mathfrak m^\\vee$ is distinguished, so the two hypotheses are equivalent. We define $S_{\\mathfrak m}(Δ)=\\sum_{t\\ge0}(-1)^t\\mathfrak m(ν^tΔ)$ and show that $\\mathfrak m$ is of Speh type if and only if $S_{\\mathfrak m}(Δ)\\ge0$ for every segment $Δ$. Using rigidity properties, we obtain a peeling theorem and a numerical tameness condition, and prove both hypotheses for every tame multisegment, extending the previously known classes. Finally, we construct a five-segment multisegment that is not distinguished, although every standard order with non-increasing endpoints admits a non-trivial relevant decomposition. Thus, endpoint-ordered witnesses alone cannot prove the hypothesis in general. The remaining case reduces to multisegments for which both $\\mathfrak m$ and $\\mathfrak m^\\vee$ are non-tame.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21384",
  "author_total": 1,
  "authors": [
    {
      "name": "Hariom Sharma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctf719ccdwa6q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21384",
  "fulltext_url": null,
  "id": "ctf719ccdwa6q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21384",
  "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",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7213884659",
  "pdf_url": "https://arxiv.org/pdf/2609.21384",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctf719ccdwa6q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21384",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21384"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Duality, rigidity, and peeling for multisegments: on hypotheses of Mitra, Offen, and Sayag",
  "title_text": "Duality, rigidity, and peeling for multisegments: on hypotheses of Mitra, Offen, and Sayag",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/ctf719ccdwa6q",
  "venue": "arXiv",
  "versions": []
}