  We revisit the problem that relevant parts of bandstructures for a given cell
choice can reflect exact or approximate higher symmetries of subsystems in the
cell and can therefore be significantly simplified by an unfolding procedure
that recovers the higher symmetry. We show that bandstructure unfolding can be
understood as projection onto induced irreducible representations of a group
obtained by extending the original group of translations with a number of
additional symmetry operations. The resulting framework allows us to define a
generalized unfolding procedure which includes the point group operations and
can be applied to any quantity in the reciprocal space. The unfolding of the
Brillouin zone follows naturally from the properties of the induced irreducible
representations. In this context, we also introduce a procedure to derive
tight-binding models of reduced dimensionality by making use of point group
symmetries. Further, we show that careful consideration of unfolding has
important consequences on the interpretation of angle resolved photoemission
experiments. Finally, we apply the unfolding procedure to various
representative examples of Fe-based superconductor compounds and show that the
one iron picture arises as an irreducible representation of the glide mirror
group and we comment on the consequences for the interpretation of one-iron
versus two-iron Brillouin zone representations.
