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 arithm etic classes in dimension three, it yields, respectively, 33, 24, 16 linear representations of dimension 6, 8, 12. We give the corresponding Molien fu nctions. 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 re sults Ire presented in the form of short tables. The comparison with the ta ble of the previous chapter is instructive. Using the possibility to make p lots of invariant function for the two-dimensional crystallography we explo it our corresponding results and also study the orbit spaces. (C) 2001 Else vier Science B.V. All rights reserved.