On the homology of partial group representations

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 $G$.