  In this work, we explore Partitioned Independent Component Analysis (PICA),
an extension of the well-established Independent Component Analysis (ICA)
framework. Traditionally, ICA focuses on extracting a vector of independent
source signals from a linear combination of them defined by a mixing matrix. We
aim to provide a comprehensive understanding of the identifiability of this
mixing matrix in ICA. Significant to our investigation, recent developments by
Mesters and Zwiernik relax these strict independence requirements, studying the
identifiability of the mixing matrix from zero restrictions on cumulant
tensors. In this paper, we assume alternative independence conditions, in
particular, the PICA case, where only partitions of the sources are mutually
independent. We study this case from an algebraic perspective, and our primary
result generalizes previous results on the identifiability of the mixing
matrix.
