Configuration spaces and peak representations

TL;DR

This paper explores the cohomological interpretation of peak representations in symmetric and hyperoctahedral groups, linking algebraic idempotents to topological configuration spaces and providing new structural insights.

Contribution

It introduces a cohomological interpretation for peak algebra representations and relates them to higher Lie characters and Jordan brackets.

Findings

Peak representations are sums of Thrall's higher Lie characters

Provides Hilbert series and branching rule recursions

Establishes a connection to Jordan brackets

Abstract

Within the group algebras of the symmetric and hyperoctahedral groups, one has their descent algebras and families of Eulerian idempotents. These idempotents are known to generate group representations with topological interpretations, as the cohomology of configuration spaces of types A and B. We provide an analogous cohomological interpretation for the representations generated by idempotents in the peak algebra, called the peak representations. We describe the peak representations as sums of Thrall's higher Lie characters, give Hilbert series and branching rule recursions for them, and discuss a connection to Jordan brackets.