Derived character maps of group representations

Berest, Yuri; Ramadoss, Ajay C

doi:10.2140/agt.2024.24.4991

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Derived character maps of group representations

Yuri Berest and Ajay C Ramadoss

Algebraic & Geometric Topology 24 (2024) 4991–5044

DOI: 10.2140/agt.2024.24.4991

Abstract

We define and study (derived) character maps of finite-dimensional representations of $\infty$ –groups. As models for $\infty$ –groups we take homotopy simplicial groups, ie the homotopy simplicial algebras over the algebraic theory of groups (in the sense of Badzioch (2002)). We introduce cyclic, symmetric and representation homology for “group algebras” $k [Γ]$ of such groups and construct canonical trace maps (natural transformations) relating these homology theories. We show that, in the case of one-dimensional representations, our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces known in stable homotopy theory. Using this topological interpretation, we deduce some algebraic results on 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 stable behavior of the derived character maps of $n$ –dimensional representations as $n \to \infty$ , in which case we show that these maps “converge” to become isomorphisms.

Keywords

representation variety, cyclic homology, symmetric homology, simplicial group, Dold–Thom space, homotopy colimit, Goodwillie calculus

Mathematical Subject Classification

Primary: 18A25, 18G15, 19D55, 55N35

Secondary: 14A30, 55P42

References

Publication

Received: 2 January 2023

Revised: 7 November 2023

Accepted: 23 November 2023

Published: 27 December 2024

Authors

Yuri Berest
Department of Mathematics Cornell University Ithaca, NY United States

Ajay C Ramadoss
Department of Mathematics Indiana University Bloomington, IN United States

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Volume 24, issue 9 (2024)