{
  "id": "2506.21525",
  "version": "v1",
  "published": "2025-06-26T17:45:30.000Z",
  "updated": "2025-06-26T17:45:30.000Z",
  "title": "The spectrum of global representations for families of bounded rank and VI-modules",
  "authors": [
    "Miguel Barrero",
    "Tobias Barthel",
    "Luca Pol",
    "Neil Strickland",
    "Jordan Williamson"
  ],
  "comment": "72 pages; all comments welcome",
  "categories": [
    "math.RT",
    "math.AT",
    "math.CT"
  ],
  "abstract": "A global representation is a compatible collection of representations of the outer automorphism groups of the finite groups belonging to a family $\\mathscr{U}$. These arise in classical representation theory, in the study of representation stability, as well as in global homotopy theory. In this paper we begin a systematic study of the derived category $\\mathsf{D}(\\mathscr{U};k)$ of global representations over fields $k$ of characteristic zero, from the point-of-view of tensor-triangular geometry. We calculate its Balmer spectrum for various infinite families of finite groups including elementary abelian $p$-groups, cyclic groups, and finite abelian $p$-groups of bounded rank. We then deduce that the Balmer spectrum associated to the family of finite abelian $p$-groups has infinite Krull dimension and infinite Cantor--Bendixson rank, illustrating the complex phenomena we encounter. As a concrete application, we provide a complete tt-theoretic classification of finitely generated derived VI-modules. Our proofs rely on subtle information about the growth behaviour of global representations studied in a companion paper, as well as novel methods from non-rigid tt-geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-26T17:45:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global representation",
      "bounded rank",
      "vi-modules",
      "finite groups",
      "finite abelian"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 72,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}