  A common approach for analyzing hypergraphs is to consider the projected
adjacency or Laplacian matrices for each order of interactions (e.g., dyadic,
triadic, etc.). However, this method can lose information about the hypergraph
structure and is not universally applicable for studying dynamical processes on
hypergraphs, which we demonstrate through the framework of cluster
synchronization. Specifically, we show that the projected network does not
always correspond to a unique hypergraph structure. This means the projection
does not always properly predict the true dynamics unfolding on the hypergraph.
Additionally, we show that the symmetry group consisting of permutations that
preserve the hypergraph structure can be distinct from the symmetry group of
its projected matrix. Thus, considering the full hypergraph is required for
analyzing the most general types of dynamics on hypergraphs. We show that a
formulation based on node clusters and the corresponding edge clusters induced
by the node partitioning, enables the analysis of admissible patterns of
cluster synchronization and their effective dynamics. Additionally, we show
that the coupling matrix projections corresponding to each edge cluster
synchronization pattern, and not just to each order of interactions, are
necessary for understanding the structure of the Jacobian matrix and performing
the linear stability calculations efficiently.
