Restricted Invariants on the Space of Elasticity Tensors

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.