  Here, the structural symmetries of a hypergraph are represented through
equivalence relations on the vertex set of the hypergraph.
  A matrix associated with the hypergraph may not reflect a specific structural
symmetry. In the context of a given symmetry within a hypergraph, we
investigate a collection of matrices that encapsulate information about the
symmetry. Our investigation reveals that certain structural symmetries in a
hypergraph manifest observable effects on the eigenvalues and eigenvectors of
designated matrices associated with the hypergraph.
  We identify invariant subspaces of these specific matrices as a consequence
of symmetries present in the hypergraph. These invariant subspaces elucidate
analogous behaviours observed in certain clusters of vertices during random
walks and other dynamical processes on the hypergraph.
