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Derived Character Maps of Groups Representations

Yuri Berest, Ajay C. Ramadoss

TL;DR

The paper develops a comprehensive framework for derived character maps from cyclic to representation-theoretic invariants of ∞-groups modeled by homotopy simplicial groups. It introduces representation and cyclic/symmetric homology for these objects, constructs derived character maps, and provides a topological realization via loop spaces and Dold–Thom spaces. A key achievement is the stable isomorphism Λ_k[\overline{HC}_*(k[Γ])] ≅ HR_*(Γ,GL_∞)^GL_∞ in characteristic zero, tying string-topology–type invariants to stabilized representation theory and Poisson geometry. The work also employs Goodwillie calculus and operad technology to illuminate the CS and SR maps, yielding nonlinear extensions that connect stable homotopy data with derived Poisson brackets on geometric models such as simply-connected manifolds.

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.

Derived Character Maps of Groups Representations

TL;DR

The paper develops a comprehensive framework for derived character maps from cyclic to representation-theoretic invariants of ∞-groups modeled by homotopy simplicial groups. It introduces representation and cyclic/symmetric homology for these objects, constructs derived character maps, and provides a topological realization via loop spaces and Dold–Thom spaces. A key achievement is the stable isomorphism Λ_k[\overline{HC}_*(k[Γ])] ≅ HR_*(Γ,GL_∞)^GL_∞ in characteristic zero, tying string-topology–type invariants to stabilized representation theory and Poisson geometry. The work also employs Goodwillie calculus and operad technology to illuminate the CS and SR maps, yielding nonlinear extensions that connect stable homotopy data with derived Poisson brackets on geometric models such as simply-connected manifolds.

Abstract

In this paper, we construct and study derived character maps of finite-dimensional representations of -groups. As models for -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 is a field of characteristic zero. We also study the behavior of the derived character maps of -dimensional representations in the stable limit as , in which case we show that they `converge' to become isomorphisms.
Paper Structure (23 sections, 33 theorems, 236 equations)

This paper contains 23 sections, 33 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Shapiro Lemma for Model Categories
  3. Notation and conventions
  4. Homotopy colimits
  5. Homotopy coends
  6. Representation and Cyclic Homology of Homotopy Simplicial Groups
  7. Homotopy simplicial groups
  8. Representation homology
  9. Cyclic homology
  10. Derived character maps
  11. Topological Realization of Derived Character Maps
  12. The space $X_{\Gamma}$
  13. Symmetric homology
  14. Symmetric homology vs representation homology
  15. Generalization to monoids
  16. ...and 8 more sections

Key Result

theorem 1.1

For any homotopy simplicial group $\Gamma$, there is a natural isomorphism where $\,\mathrm{SP}^\infty\! (B\Gamma)\,$ denotes the classical Dold-Thom space of the classifying space of $\Gamma\,$.

Theorems & Definitions (70)

  • theorem 1.1
  • theorem 1.2
  • corollary 1.1
  • theorem 1.3
  • theorem 2.1: see Cis
  • remark 2.1
  • theorem 2.2: Cofinality Theorem
  • lemma 2.1
  • proof
  • lemma 2.2: Quillen
  • ...and 60 more