Irreducible Cartesian Tensors

Abstract

This paper considers certain simple and practically useful properties of Cartesian tensors in three-dimensional space which are irreducible under the three-dimensional rotation group. Ordinary tensor algebra is emphasized throughout and particular use is made of natural tensors having the least rank consistent with belonging to a particular irreducible representation of the rotation group. An arbitrary tensor of rank n may be reduced by first deriving from the tensor all its linearly independent tensors in natural form, and then by embedding these lower-rank tensors in the tensor space of rank n. An explicit reduction of third-rank tensors is given as well as a convenient specification of fourth- and fifth-rank isotropic tensors. A particular classification of the natural tensors is through a Cartesian parentage scheme, which is developed. Some applications of isotropic tensors are given.