Hyperoctahedral groups, wreath products, and a general Weyl's recipe

Abstract

The permutation-inversion structure of the hyperoctahedral group is presented in the light of the general Weyl's recipe, associated with groups of automorphisms. The hyperoctahedral group is formally expressed as a wreath product. It is shown that the hidden symmetry revealed by application of this recipe can be interpreted crystallographically in terms of Bravais cells. This interpretation, together with the periodic Born-Karman conditions, suggests some generalizations of crystal symmetry groups similar to symmetry groups of non-rigid (floppy) molecules. It is shown that cubic crystals with the range of flopping limited to two lattice constants can have symmetry elements of order seven, whereas elements of order five are still forbidden.