  In recent years, there have been rapid advances in the parallel fields of
electronic and photonic topological crystals. Topological photonic crystals in
particular show promise for coherent transport of light and quantum information
at macroscopic scales. In this work, we apply for the first time the recently
developed theory of "Topological quantum chemistry" to the study of band
structures in photonic crystals. This method allows us to design and diagnose
topological photonic band structures using only group theory and linear
algebra. As an example, we focus on a family of crystals formed by elliptical
rods in a triangular lattice. We show that the symmetry of Bloch states in the
Brillouin zone can determine the position of the localized photonic wave
packets describing groups of bands. By modifying the crystal structure and
inverting bands, we show how the centers of these wave packets can be moved
between different positions in the unit cell. Finally, we show that for shapes
of dielectric rods, there exist isolated topological bands which do not admit a
well-localized description, representing the first physical instance of
"fragile" topology in a truly noninteracting system. Our work demonstrates how
photonic crystals are the natural platform for the future experimental
investigation of fragile topological bands.
