Derived Character Maps of Groups Representations
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913629739155456 |
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| author | Berest, Yuri Ramadoss, Ajay C. |
| author_facet | Berest, Yuri Ramadoss, Ajay C. |
| contents | In this paper, we construct and study derived character maps of finite-dimensional representations of $\infty$-groups. As models for $\infty$-groups we take homotopy simplicial groups, i.e. homotopy simplicial algebras over the algebraic theory of groups (in the sense of Badzioch). We define cyclic, symmetric and representation homology for `group algebras' over such groups and construct canonical trace maps relating these homology theories. In the case of one-dimensional representations, we show that our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces that are well studied in homotopy theory. Using this topological interpretation, we deduce some algebraic results about representation homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring $k$ is a field of characteristic zero. We also study the behavior of the derived character maps of $n$-dimensional representations in the stable limit as $ n\to \infty$, in which case we show that they `converge' to become isomorphisms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_01304 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Derived Character Maps of Groups Representations Berest, Yuri Ramadoss, Ajay C. Algebraic Topology Category Theory K-Theory and Homology Representation Theory In this paper, we construct and study derived character maps of finite-dimensional representations of $\infty$-groups. As models for $\infty$-groups we take homotopy simplicial groups, i.e. homotopy simplicial algebras over the algebraic theory of groups (in the sense of Badzioch). We define cyclic, symmetric and representation homology for `group algebras' over such groups and construct canonical trace maps relating these homology theories. In the case of one-dimensional representations, we show that our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces that are well studied in homotopy theory. Using this topological interpretation, we deduce some algebraic results about representation homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring $k$ is a field of characteristic zero. We also study the behavior of the derived character maps of $n$-dimensional representations in the stable limit as $ n\to \infty$, in which case we show that they `converge' to become isomorphisms. |
| title | Derived Character Maps of Groups Representations |
| topic | Algebraic Topology Category Theory K-Theory and Homology Representation Theory |
| url | https://arxiv.org/abs/2210.01304 |