What happens to the space group of a crystal when introducing a non-trivial basis? I am trying to understand crystallography and the space groups of crystals, but I have one major question bugging me. The book I am using adresses different lattice symmetries and applications of group theory. More specifically, the electron energy bands are characterised by the irreducible representations of the point group of the Bravais lattice.
By basis I mean the atoms associated with a lattice point.
Is this characterisation altered or does it break down in any way when a non-trivial basis is used?
I would suspect the answer to be along the lines: "The form of the bands are changed, but the characterisation is conserved...", but this is only guesswork.
 A: By adding atoms to the crystal basis, the space group will generally change and so will the symmetry classification of electronic states $|\psi_{nk}\rangle$. To be a bit more complete, let's look at exactly how states are classified and labeled.
If we have a symmorphic space group $\mathbb{S}$, then for a given $k$ we can define its little cogroup $\mathbb{G}_k$ (group of the wavevector) by
$$ \mathbb{G}_k = \{\:(R|0)\in \mathbb{S} \:\:\:|\:\:\: Rk\equiv k \:\} $$
where $Rk\equiv k$ means equivalence up to a reciprocal lattice vector. In a crystal, electronic states $|\psi_{nk}\rangle$ are classified by the irreducible representations of the little cogroup of $k$. Each high symmetry line in the Brillouin Zone has a single little cogroup associated with it and the irreps will then label the bands along that line.
Example
FCC with one atom basis
The basic FCC lattic has space group $Fm\bar{3}m$ (#225). If we look at the high symmetry line $\Delta:\Gamma\to X$, we can find (from this source) that the little cogroup is $4mm$ ($C_{4v}$). Then looking at the irreps here, we see there are 5 of them including one 2D irrep
FCC with two atom basis
A zincblende crystal is FCC but has a two atom basis (like CdTe). It has space group $F\bar{4}3m$ (#216). Looking at the same high symmetry line $\Delta:\Gamma\to X$, the little cogroup in this case is $2mm$ ($C_{2v}$) (as found here). $C_{2v}$ has only 4 irreps and they are all 1D. So the symmetry classification is clearly different.
