= Solution
For a local electric-dipole response of order $n$, write
$$
P_i(\omega_s)=\epsilon_0\sum_{j_1\ldots j_n}\chi^{(n)}_{ij_1\ldots j_n}(\omega_s;\omega_1,\ldots,\omega_n)
E_{j_1}(\omega_1)\cdots E_{j_n}(\omega_n),\qquad
\omega_s=\sum_r\omega_r.
$$
Its susceptibility is a rank-$(n+1)$ <tensor>. Initially it has $3^{n+1}$ components in three dimensions, before symmetries and <frequency> constraints are applied. <nonlinear-susceptibility tensor symmetry> involves several logically different restrictions.
<Neumann's principle> requires invariance under the material <point group>. For an operation $R$,
$$
\chi_{ij_1\ldots j_n}=R_{ia}R_{j_1b_1}\cdots R_{j_nb_n}\chi_{ab_1\ldots b_n}.
$$
For inversion, $R=-I$, so $\chi^{(n)}=(-1)^{n+1}\chi^{(n)}$. Thus \b[all even orders vanish in a centrosymmetric material's local bulk electric-dipole response]. In particular $\chi^{(2)}=0$ there, while $\chi^{(3)}$ is permitted. Surface symmetry breaking, an applied bias or nonlocal magnetic/quadrupole response lies outside that statement. A noncentrosymmetric crystal permits some second-order components, but other point-group operations can still eliminate a chosen component.
Intrinsic permutation symmetry follows from interchanging commuting input field factors, provided the associated <frequency> labels are interchanged too. At second order,
$$
\chi^{(2)}_{ijk}(\omega_s;\omega_1,\omega_2)=\chi^{(2)}_{ikj}(\omega_s;\omega_2,\omega_1).
$$
For equal input <frequencies> this makes the last two spatial indices symmetric, reducing 27 components to 18 before material symmetries. At unequal <frequencies> it does not identify the two index arrangements with the arguments held fixed. Reality of the time-domain polarization separately gives
$$
\chi^{(n)}(-\omega_s;-\omega_1,\ldots,-\omega_n)=\chi^{(n)}(\omega_s;\omega_1,\ldots,\omega_n)^*.
$$
<causality> imposes dispersion relations; neither arbitrary real coefficients nor full index symmetry should be inferred merely from the word lossless.
<Kleinman symmetry> is a stronger approximation far from relevant resonances when <frequency> dispersion can be neglected. A frequency-independent effective interaction <energy> makes its derivatives interchangeable, allowing full permutation of output and input spatial indices. At second order a fully symmetric rank-three <tensor> has $\binom{5}{3}=10$ components. Close to resonance or across a dispersive band, that reduction can fail even when <absorption> at the measured <frequencies> is small. The original analysis distinguishes high-resonance electronic response from strongly dispersive lattice mechanisms and identifies additional symmetry constraints on the former.<Nonlinear Dielectric Polarization in Optical Media by D. A. Kleinman>[abstract, mechanism dependence and additional symmetry requirements]{ref}
For an <isotropic fourth-rank susceptibility>,
$$
\chi^{(3)}_{ijkl}=a\delta_{ij}\delta_{kl}+b\delta_{ik}\delta_{jl}+c\delta_{il}\delta_{jk}.
$$
If the relevant input slots are fully interchangeable, the coefficients coincide and \b[$\chi_{xxxx}^{(3)}=3\chi_{xxyy}^{(3)}$]. With distinct or mixed-sign <frequency> slots, <frequency> arguments must be followed and isotropy alone leaves the three displayed structures. These distinctions explain why <tensor> contractions, pump polarizations and operating <frequencies> must be specified before comparing effective self- and cross-phase coefficients.
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