Cluster synchronization (a type of synchronization where different groups of nodes in the system of coupled oscillators follow distinct synchronized trajectories) on networks is a broadly analyzed phenomenon characterizing behaviors with wide areas of applicability from neuroscience to consensus dynamics. Analyzing cluster synchronization can be used to understand phenomena such as remote synchronization and the emergence of chimera states. Ideas from graph and equivariant dynamical systems theory can be applied to deduce admissible patterns of synchronization and simplify their stability analysis. However, higher order interactions may be required to describe many social, biological, and ecological systems, making it necessary to go beyond the pairwise interaction analysis to study certain phenomena such as consensus dynamics, epidemic spreading, and metabolic reactions. While complete synchronization and its stability have been analyzed very recently for such systems, and examples from consensus dynamics, where the system settles on a fixed point, have been analyzed, general cluster synchronization has not been considered. To address that, we formulate conditions for cluster synchronization based on the hypergraph structure from (external) equitable partition and symmetry perspective. Then, we show how to reduce the dimensionality of stability calculation based on the hypergraph structure for any specific pattern of cluster synchronization. A specific example of an admissible pattern of synchronization and its analysis is shown in Figure 1. Our results are an extension of existing cluster synchronization literature to higher order systems and could be of interest to a larger audience.
