{
  "abstract": "Finite-dimensional zigzag modules decompose into interval modules. For an ordinary persistence module, every fully supported interval submodule of a restriction splits as a direct summand, whereas this need not hold for a zigzag module. We study this failure of splitting. A breaker pair is a local configuration of two overlapping interval summands with a backward arrow at the left boundary of the second interval and a forward arrow at the right boundary of the first. A single breaker pair gives a local nonsplitting interval submodule, but it need not control a prescribed larger restriction. We therefore introduce breaker chains. Our main result shows that a restriction $V[p,q]$ contains a nonsplitting submodule isomorphic to $I[p,q]$ if and only if its interval decomposition contains a breaker chain spanning $[p,q]$. Equivalently, the proper interval summands $J$ for which $\\operatorname{Hom}(I[p,q],J)\\ne0$ have supports covering $[p,q]$. This cover formulation gives a constructive description of nonsplitting embeddings, a trichotomy for their splitting behavior, and an $O(m\\log m)$ algorithm from an oriented barcode with $m$ intervals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14664",
  "author_total": 2,
  "authors": [
    {
      "name": "Krishnendu Gongopadhyay"
    },
    {
      "name": "Ramveer Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bn6d45xyqca26.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14664",
  "fulltext_url": null,
  "id": "bn6d45xyqca26",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14664",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7213281257",
  "pdf_url": "https://arxiv.org/pdf/2609.14664",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bn6d45xyqca26.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14664",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14664"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Breaker Chains and Splitting of Interval Submodules in Zigzag Modules",
  "title_text": "Breaker Chains and Splitting of Interval Submodules in Zigzag Modules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/bn6d45xyqca26",
  "venue": "arXiv",
  "versions": []
}