{
  "id": "2407.20405",
  "version": "v1",
  "published": "2024-07-29T20:17:01.000Z",
  "updated": "2024-07-29T20:17:01.000Z",
  "title": "Rank and symmetries of signature tensors",
  "authors": [
    "Francesco Galuppi",
    "Pierpaola Santarsiero"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. In this paper we propose a systematic study of basic properties of signature tensors, starting from their rank, symmetries and conciseness. We prove a sharp upper bound on the rank of signature tensors of piecewise linear paths. We show that there are no skew-symmetric signature tensors of order three or more, and we also prove that specific instances of partial symmetry can only happen for tensors of order three. Finally, we give a simple geometric characterization of paths whose signature tensors are not concise.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-29T20:17:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N07",
      "15A69",
      "15A72",
      "60L10"
    ],
    "keywords": [
      "sharp upper bound",
      "skew-symmetric signature tensors",
      "simple geometric characterization",
      "systematic study",
      "basic properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}