It is shown that in the direct product of two wave vector stars k?k′, if k"∈k?k′, the wave vector group, Gk", may belong to any one of the following cases: A) Both Gk and Gk′ are subgroups of Gk"; B) Gk=Gk′= Gk"; C) Gk" = Gs; D) Gs is the subgroup of Gk". The reduction of an integral A=μ″i″(k″)|fμi(k)|fμ′i′(k′)>, is then studied for each of the four cases. It is found that for all cases, the value of the integral can be expressed in terms of the matrix element of a transformed matrix U by which the reducible representation has been reduced. The corresponding reducible representations Г′s are respec-tively: A) (Гki?Гk′i′)(?);B)Гki?Гk′i′;C)Гki(s)?Гk′i′(s); and D)(Гki(s)?Гk′i′(s))(?). The final form of the integral is then obtained by finding the explicit form of the transformation matrix by using the pseudo-projection operator.
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