Decomposition of direct product at an arbitrary Brillouin zone point:   $D^{(\bigstar{R})(m)}$ $\otimes$ $D^{(\bigstar{-R})(m)}$
  A general rule is presented for the decomposition of the direct product of irreducible representation at arbitrary Brillouin zone point $\bf{R}$ with its negative: the number of the appearences of the zone center representation equals the dimensionality of the representation. This rule is applicable for all space groups. Although in most situations the interesting physics takes place at high symmetry points in the Brillouin zone, this general rule is useful for situations where double excitations are considered. It is shown that double excitations from arbitrary Brillouin point $\bf{R}$ have the right symmetry to participate in all optical experiments regardless of polarization directions. 
Jian Li, Jiufeng J. Tu, Joseph L. Birman
cond-mat.mtrl-sci