  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 .
