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A general rule is presented for the decomposition of the direct product ofirreducible representation at arbitrary Brillouin zone point $\bf{R}$ with itsnegative: the number of the appearences of the zone center representationequals the dimensionality of the representation. This rule is applicable forall space groups. Although in most situations the interesting physics takesplace at high symmetry points in the Brillouin zone, this general rule isuseful for situations where double excitations are considered. It is shown thatdouble excitations from arbitrary Brillouin point $\bf{R}$ have the rightsymmetry to participate in all optical experiments regardless of polarizationdirections.

Decomposition of direct product at an arbitrary Brillouin zone point: $D^{(\bigstar{R})(m)}$ $\otimes$ $D^{(\bigstar{-R})(m)}$