The ring of invariant real functions on the Brillouin zone

With the coordinates chosen in the previous chapter, we show explicitly how to linearize the action of crystallographic space groups on the Brillouin zone. For two-dimensional crystallography it yields eight four-dimensional representations and five six-dimensional representations. For the 73 arithmetic classes in dimension three, it yields, respectively, 33, 24, 16 linear representations of dimension 6, 8, 12. We give the corresponding Molien functions. For the representations of dimensions four and six, we compute the invariants (up to 96 numerator invariants for the R lattices). We can even extend the results to the 16 hexagonal arithmetic classes. All obtained results are presented in the form of short tables. The comparison with the table of the previous chapter is instructive. Using the possibility to make plots of invariant function for the two-dimensional crystallography we exploit our corresponding results and also study the orbit spaces.