Irreducible tensors third order

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

The task of expressing K in terms of irreducible tensors is somewhat more involved. The general problem of reducing tensors of any order into irreducible components is discussed by Coope et al.,3 which was referred to earlier. These authors show that a general third-order tensor can always be expressed in the form... 

Here, y is the unique symmetric and traceless (irreducible) part of Q, whereas D(i) are symmetric and traceless second-order (irreducible) tensors, and v(i) and S are vectors and a scalar. It may be noted that, if we write a general third-order tensor in the form (8 33), there is no loss of generality in assuming that the second-order tensor components are symmetric. The antisymmetric part of any second-order tensor can always be represented by a vector. For example, if D = D,s + D" then the antisymmetric part can always be written as D"= e d where d = — e D" and included in the vector terms of (8-33). 

The specific case of the third-order tensor K = Wu.x, was considered by Nadim and Stone.4 In index notation, the irreducible description of K is... 

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