Spectral decomposition of hypergraph automorphism compatible matrices

This study explores the relationship between hypergraph automorphisms and the spectral properties of matrices associated with hypergraphs. For an automorphism $f$, an $f$-compatible matrices capture aspects of the symmetry, represented by $f$, within the hypergraph. First, we explore rotation, a specific kind of automorphism and find that the spectrum of any matrix compatible with a rotation can be decomposed into the spectra of smaller matrices associated with that rotation. We show that the spectrum of any $f$-compatible matrix can be decomposed into the spectra of smaller matrices associated with the component rotations comprising $f$. Further, we study a hypergraph symmetry termed unit-automorphism, which induces bijections on the hyperedges, though not necessarily on the vertex set. We show that unit automorphisms also lead to the spectral decomposition of compatible matrices.