Global representation theory: Homological foundations
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910235415805952 |
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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 groups belonging to some collection of finite groups $\mathscr{U}$. Global representations assemble into an abelian category $\mathsf{A}(\mathscr{U})$, simultaneously generalising classical representation theory and the category of VI-modules appearing in the representation theory of the general linear groups. In this paper we establish homological foundations of its derived category $\mathsf{D}(\mathscr{U})$. We prove that any complex of projective global representations is DG-projective, and hence conclude that the derived category admits an explicit model as the homotopy category of projective global representations. We show that from a tensor-triangular perspective it exhibits some unusual features: for example, there are very few dualizable objects and in general many more compact objects. Under more restrictive conditions on the family $\mathscr{U}$, we then construct torsion-free classes for global representations which encode certain growth properties in $\mathscr{U}$. This lays the foundations for a detailed study of the tensor-triangular geometry of derived global representations which we pursue in forthcoming work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global representation theory: Homological foundations 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 groups belonging to some collection of finite groups $\mathscr{U}$. Global representations assemble into an abelian category $\mathsf{A}(\mathscr{U})$, simultaneously generalising classical representation theory and the category of VI-modules appearing in the representation theory of the general linear groups. In this paper we establish homological foundations of its derived category $\mathsf{D}(\mathscr{U})$. We prove that any complex of projective global representations is DG-projective, and hence conclude that the derived category admits an explicit model as the homotopy category of projective global representations. We show that from a tensor-triangular perspective it exhibits some unusual features: for example, there are very few dualizable objects and in general many more compact objects. Under more restrictive conditions on the family $\mathscr{U}$, we then construct torsion-free classes for global representations which encode certain growth properties in $\mathscr{U}$. This lays the foundations for a detailed study of the tensor-triangular geometry of derived global representations which we pursue in forthcoming work. |
| title | Global representation theory: Homological foundations |
| topic | Representation Theory Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2505.21449 |