The spectrum of global representations for families of bounded rank and VI-modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913914365673472 |
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| author | Barrero, Miguel Barthel, Tobias Pol, Luca Strickland, Neil Williamson, Jordan |
| author_facet | Barrero, Miguel Barthel, Tobias Pol, Luca Strickland, Neil Williamson, Jordan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21525 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The spectrum of global representations for families of bounded rank and VI-modules Barrero, Miguel Barthel, Tobias Pol, Luca Strickland, Neil Williamson, Jordan Representation Theory Algebraic Topology Category Theory 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. |
| title | The spectrum of global representations for families of bounded rank and VI-modules |
| topic | Representation Theory Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2506.21525 |