Derived character maps of group representations

Berest, Yuri¹ ; Ramadoss, Ajay C²

¹ Department of Mathematics, Cornell University, Ithaca, NY, United States
² Department of Mathematics, Indiana University, Bloomington, IN, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 9, pp. 4991-5044

Résumé

We define and study (derived) character maps of finite-dimensional representations of ∞–groups. As models for ∞–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 →∞, in which case we show that these maps “converge” to become isomorphisms.

Détail

DOI : 10.2140/agt.2024.24.4991

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