  Full synchronization of dynamical elements coupled via hypergraphs can be
analyzed with the hypergraph projection onto dyadic matrices, but this is not
sufficient for analyzing cluster synchronization. Here we develop the necessary
formalism. We introduce the notion of edge clusters and show how node and edge
partitions allow us to verify admissible states and simplify their linear
stability calculations. This provides a principled way to track dynamics on
hypergraphs, and the projected Laplacian matrices based on each edge cluster
are essential to linear stability analysis and its dimensionality reduction.
This work goes beyond full synchronization and beyond dyadic interactions.
