= Solution
For a second-order response, $P_i^{(2)}(\omega_3)=\epsilon_0\sum_{jk}\chi_{ijk}^{(2)}(\omega_3;\omega_1,\omega_2)E_j(\omega_1)E_k(\omega_2)$, with the appropriate <frequency> conservation and negative-frequency convention for difference-frequency processes. Interchanging the two input field-frequency pairs gives <intrinsic permutation symmetry of nonlinear susceptibility>:
$$
\chi_{ijk}^{(2)}(\omega_3;\omega_1,\omega_2)
=\chi_{ikj}^{(2)}(\omega_3;\omega_2,\omega_1).
$$
For <second-harmonic generation> the input frequencies agree, so the last two Cartesian indices are symmetric and can be contracted into six input-pair labels. This reduces the unrestricted 27 Cartesian entries to at most 18 before crystal symmetries are imposed.
Crystal point-group transformations must leave the material tensor invariant and can <force> entries to vanish or become related. Under spatial inversion both <electric field> and <polarization> reverse, whereas the field product is even. A centrosymmetric bulk therefore has zero electric-dipole $\chi^{(2)}$. Surfaces and interfaces can break <inversion symmetry>, and higher-multipole processes are outside that simple prohibition.
<Kleinman symmetry> is a stronger approximate permutation symmetry used sufficiently far from resonances when dispersion can be neglected over the participating frequencies. It is not justified merely because two input frequencies are equal; resonant or appreciably dispersive media need the full frequency-dependent relations. These distinctions determine which <polarization> and propagation geometries have nonzero effective coupling, in addition to whether they satisfy <phase matching>.<Nonlinear Dielectric Polarization in Optical Media by D. A. Kleinman>[1962, symmetry of the nonlinear dielectric response]{ref}
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