{
  "abstract": "Every square matrix is uniquely the sum of a symmetric matrix and a skew-symmetric matrix. We extend this familiar fact to higher-order tensors: every cubic $k$-tensor is uniquely the sum of an $m$-piecewise symmetric tensor and a $(k-m)$-piecewise skew-symmetric tensor, for each choice of $m\\leq k$. We study these tensor spaces via the combinatorics of word descents and compositions. Under Schur--Weyl duality, the two spaces naturally decompose into descent representations of the symmetric group. In fact, we prove a variant of Solomon's classical decomposition of its group algebra that explains this decomposition. Our motivation stems from signature tensors in stochastic analysis, algebraic geometry and data science: more specifically, we show that our tensor space decompositions determine the vanishing ideals of the signatures of piecewise linear paths with a fixed number of segments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04712",
  "author_total": 2,
  "authors": [
    {
      "name": "Felix Lotter"
    },
    {
      "name": "Rosa Preiß"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2pfrjfbqskben.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04712",
  "fulltext_url": null,
  "id": "2pfrjfbqskben",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04712",
  "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.PR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7167606243",
  "pdf_url": "https://arxiv.org/pdf/2607.04712",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2pfrjfbqskben.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2607.04712",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04712"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Piecewise Symmetric Tensors",
  "title_text": "Piecewise Symmetric Tensors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/2pfrjfbqskben",
  "venue": "arXiv",
  "versions": []
}