How to use the momentum matrix of the reduced k-points

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I have a set of k-points, e.g. k1,k2,k3,k4. and they are equivalent by symmetry.

Now I have calculated the momentum matrix element <i|p|j> at k1 point ONLY,

and then calculate the optical properties which, for example, depend on <i|p|j><j|p|i>

I have to make a summation on four k-points (k1,k2,k3,k4) for dk integration.

So, how to make this summation only on the basis of <i|p|j> of k1 point?

Can I simply do by multiplying <i|p|j> by the weight of k1 point? because these four k-points are equivalent.

If not, could you give me some references?Any help will be appreciated.

Youzhao Lan
 
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I am no specialist on this, but I think that the p and |i> span a representation of the point group which is also responsible for the equivalence of the k points. So you have to find the irreducible representations spanned by p and |i> and find those combinations of irreducible representations formed from <j|, p and |i> that the total matrix element <j|p|i> is totally symmetric.