{
  "id": "2407.19132",
  "version": "v1",
  "published": "2024-07-27T00:32:47.000Z",
  "updated": "2024-07-27T00:32:47.000Z",
  "title": "Tensor spaces and the geometry of polynomial representations",
  "authors": [
    "Nate Harman",
    "Andrew Snowden"
  ],
  "comment": "20 pages",
  "categories": [
    "math.RT",
    "math.AG",
    "math.LO"
  ],
  "abstract": "A \"tensor space\" is a vector space equipped with a finite collection of multi-linear forms. In previous work, we showed that (for each signature) there exists a universal homogeneous tensor space, which is unique up to isomorphism. Here we generalize that result: we show that each Zariski class of tensor spaces contains a weakly homogeneous space, which is unique up to isomorphism; here, we say that two tensor spaces are \"Zariski equivalent\" if they satisfy the same polynomial identities. Our work relies on the theory of $\\mathbf{GL}$-varieties developed by Bik, Draisma, Eggermont, and Snowden.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-27T00:32:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial representations",
      "tensor spaces contains",
      "universal homogeneous tensor space",
      "polynomial identities",
      "finite collection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}