Harmonic and Cartan decompositions are used to prove that there are eight symmetry classes of elasticity tensors. Recent results in apparent contradiction with this conclusion are discussed in a short history of the problem.

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The action of the orthogonal group on the space of plane elasticity tensors has been the subject of some recent investigations. It is shown here that the approach based on the "harmonic decomposition" technique, which is also used in a three-dimensional setting, gives a unified perspective on this issue. We construct explicit relationships between invariants and quantities derived from such an approach and what was found earlier by Tsai and Pagano and, more recently, through the "polar method" and the use of complex variables

Dedicated to Professor Michael Hayes on the occasion of his 65th birthday.Abstract: A linear function defined on the space of elasticity tensors is a restricted invariant under a group of rotations G if it has an invariant restriction to a proper subspace which is larger than the set left fixed by the action of G itself. A necessary and sufficient condition for a function to be a restricted invariant is given using concepts related with isotypic decomposition, Haar integration and G -dependence. The result is applied to characterize isotropic and transversely isotropic restricted invariants.