Wave vector substar group in reciprocal lattice space and its representation

In the paper we establish the new conception of the wave vector substar group and its representation that, in the study on translational symmetry breaking of crystal, can only consider the particular arms of wave vector star taking part in phase transition, but not all arms of wave vector star, unlike the traditional Landau theory. We show that using the new conception, we can effectively investigate the interesting physical properties of crystals associated with translational symmetry breaking. We can see that the studies on the complicated phase transitions related to reducible representations, such as phase transitions in perovskite KMnF3 multiferroics crystal and high temperature superconductor La2/3Mg1/2W1/2O3 (La4Mg3W3O18) compound, are much simplified by the new conception, and the theory of the wave vector substar group and its representation becomes a powerful mathematical tool being able to promote actively the study of various symmetry breaking phenomena of solid state crystals .