A Categorical Decomposition of $\mathbb C^{\times}$-fibered $p$-biset Functors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913078379020288 |
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| author | Coşkun, Olcay Muslumov, Ruslan |
| author_facet | Coşkun, Olcay Muslumov, Ruslan |
| contents | We generalize Bouc's construction of orthogonal idempotents in the double Burnside algebra to the setting of the double $\mathbb{C}^\times$-fibered Burnside algebra. This yields a structural decomposition of the evaluations of $\mathbb{C}^\times$-fibered biset functors on finite groups. We then construct a complete set of orthogonal idempotents in the category of $\mathbb{C}^\times$-fibered $p$-biset functors, leading to a categorical decomposition of this category into subcategories indexed by isomorphism classes of atoric $p$-groups. Furthermore, we introduce the notion of vertices for indecomposable functors and establish that the Ext-groups between simple functors with distinct vertices vanish. As an application, we describe a set containing composition factors of the monomial Burnside functor, thereby providing new insights into its structure. Additionally, we develop a technique for analyzing fibered biset functors via their underlying biset structures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_15163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Categorical Decomposition of $\mathbb C^{\times}$-fibered $p$-biset Functors Coşkun, Olcay Muslumov, Ruslan Representation Theory K-Theory and Homology We generalize Bouc's construction of orthogonal idempotents in the double Burnside algebra to the setting of the double $\mathbb{C}^\times$-fibered Burnside algebra. This yields a structural decomposition of the evaluations of $\mathbb{C}^\times$-fibered biset functors on finite groups. We then construct a complete set of orthogonal idempotents in the category of $\mathbb{C}^\times$-fibered $p$-biset functors, leading to a categorical decomposition of this category into subcategories indexed by isomorphism classes of atoric $p$-groups. Furthermore, we introduce the notion of vertices for indecomposable functors and establish that the Ext-groups between simple functors with distinct vertices vanish. As an application, we describe a set containing composition factors of the monomial Burnside functor, thereby providing new insights into its structure. Additionally, we develop a technique for analyzing fibered biset functors via their underlying biset structures. |
| title | A Categorical Decomposition of $\mathbb C^{\times}$-fibered $p$-biset Functors |
| topic | Representation Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2505.15163 |