On the homology of partial group representations

TL;DR

This paper establishes a connection between partial group (co)homology and classical group (co)homology through the concept of universal globalization, providing new tools to analyze partial representations.

Contribution

It introduces the universal globalization of partial group representations and demonstrates that partial homology and cohomology can be expressed via classical group (co)homology, including a spectral sequence and collapse results.

Findings

Partial group homology is isomorphic to classical group homology of the globalization.

A spectral sequence relates partial and classical group cohomology.

For countable groups, the spectral sequence collapses, simplifying the relationship.

Abstract

We study how the partial group (co)homology of a group $G$ with coefficient in a partial representation $M$ can be described using the usual group (co)homology. To address this, we introduce the concept of the \textit{universal globalization} $\Lambda(M)$ of a partial group representation $M$ of $G$. Our main result shows that the partial group homology $H^{\text{par}}_{\bullet}(G, M)$ is naturally isomorphic to the classical group homology $H_{\bullet}(G, \Lambda(M))$. We extend this result to the cohomological framework, obtaining a spectral sequence involving the classical group cohomology that converges to the partial group cohomology. Notably, when $G$ is countable, the spectral sequence collapses, resulting in a natural isomorphism $H^{\bullet}_{\text{par}}(G, M) \cong H^{\bullet}(G, \operatorname{Hom}_{K_{\text{par}} G}(\Lambda(K_{par}G), M))$, where $K_{par}G$ stands for the partial group algebra of $G$.