chapter
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d is a tensor given by

Thus, under the condition that the susceptibilities are real (all frequencies far from any resonance), the susceptibilities are unchanged for the simultaneous permutation of subscripts from the cartesian set {ijkl} and the corresponding subscripts from the frequency set {4123}, with the stipulation that the frequencies carry the proper sign. Note: The first frequency in the argument, that is, the generated frequency, carries a negative sign. The signs on the other frequencies must be such that the algebraic sum of all frequencies is zero. For example, –w4 + w3 + w2 + w1 = 0 implies that –w2 – w1 + w4 – w3 = 0. This is called full permutation symmetry. This symmetry generalizes to all orders.