A process of decomposition of fourth-rank tensors into parts irreducible with reference to the continuous group of rotation is presented. The transformation matrices between the Cartesian and spherical reducible and irreducible fourth-rank tensors are given and discussed. We have focused our attention to the purely dipolar fourth-rank tensor
symmetric to its last three indices and the dipole-octopole fourth-rank multipolar polarizability tensor
. The fourth-rank tensors intervene in a number of very important nonlinear optics processes like Kerr effect, intensity-dependent refractive index phenomena, four wave mixing, third harmonic generation and several other effects. Tensors of this kind are also very important in piezo-electric phenomena and in elasticity studies. Cartesian tensor index permutation is discussed as well as its influence on the tensor irreducible spectrum is studied. Several examples concerning fourth-rank tensors are given.
