  When describing complex interconnected systems, one often has to go beyond
the standard network description to account for generalized interactions. Here,
we establish a unified framework to simplify the stability analysis of cluster
synchronization patterns for a wide range of generalized networks, including
hypergraphs, multilayer networks, and temporal networks. The framework is based
on finding a simultaneous block diagonalization of the matrices encoding the
synchronization pattern and the network topology. As an application, we use
simultaneous block diagonalization to unveil an intriguing type of chimera
states that appear only in the presence of higher-order interactions. The
unified framework established here can be extended to other dynamical processes
and can facilitate the discovery of emergent phenomena in complex systems with
generalized interactions.
