Derived Character Maps of Groups Representations

TL;DR

This paper develops derived character maps for representations of $ $-groups modeled as homotopy simplicial groups, linking algebraic and topological aspects, and explores their properties and limits.

Contribution

It introduces and studies derived character maps for $ $-group representations, connecting algebraic homology theories with topological loop space maps.

Findings

Trace maps are of topological origin in one-dimensional cases.

Symmetric and one-dimensional representation homologies are isomorphic over characteristic zero fields.

Derived character maps become isomorphisms as the representation dimension tends to infinity.

Abstract

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.