Irreducible Representations
Given a symmetry of a system, its Hamiltonian can be broken into sectors corresponding to the irreducible representations, or irreps, of the symmetry group. (See Using Symmetry in Exact Diagonalization.)
IrrepComponent
A symmetry group in general has multiple irreps, and some of those irreps may be of dimension higher than one. The data structure that represents the component of an irrep of a symmetry is IrrepComponent, which has three fields: symmetry, the underlying symmetry, irrep_index, the irrep it refers to, and irrep_component, the component in a multidimensional irrep.
Irreps of Translation Symmetry
Translation symmetry forms an Abelian group. All irreps of a translation symmetry are, therefore, one-dimensional, corresponding to Fourier modes, or momentum sectors. Given a Bravais lattice, together with its generating translations, irreps can be computed simply as
\[\Phi_{\mathbf{k}} = \exp \left( - 2 \pi i \sum_{i} k_{i} x_{i} / L_{i} \right)\]
where $k_{i} \in \{0, 1, \ldots, L_{i}-1 \}$ labels the irrep, $x_{i}$ labels the translation operations in units of the generators, and $L_{i}$ is the order (i.e. period length) of the ith generator.
Also since all irreps are one-dimensional, IrrepComponent of translation symmetry must, therefore, always have irrep_component=1.
Irreps of Point Symmetry
Point symmetry, on the other hand, is not always Abelian. Unlike with translation symmetry which is abelian, LatticeTools.jl does not compute the irreps of the point group. Instead, it keeps a database of the point symmetries in two and three dimensions, and their irreps in PointSymmetryDatabase. (* This may later be replaced by IrrepDatabase.) The representation of the point operation, and their irreps are taken from the Bilbao Crystallographic Server.