Group Orthogonality Theorem

Let $\Gamma$ be a representation for a Group of Order $h$, then

$$\sum_R \Gamma_i(R)_{mn} {\Gamma_j(R)_{m'n'}}^* = {h \over \sqrt{l_i l_j}} \delta_{ij}\delta_{mm'} \delta_{nn'}.$$

The proof is nontrivial and may be found in Eyring et al. (1944).

References

Eyring, H.; Walker, J.; and Kimball, G. E. Quantum Chemistry. New York: Wiley, p. 371, 1944.