The use of the Clebsch-Gordan reduction of the Kronecker square of the typical representation in symmetry problems of crystal physics. I. Theoretical foundations

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Abstract
The theoretical background of a universal method for finite groups of calculating polynomial invariants and covariants (sets of functions transforming by non-identity representations) in the components of any tensor is considered. This includes the determination of the transformation properties of tensors, which is a question of determining linear covariants in the tensor components. The concepts of typical representation and typical variables, characteristic for an abstract group G and for the isomorphic family of groups of linear transformations imaged by G, are introduced. It is shown that the multilinear covariants in typical variables yield a prescription for the construction of tensor and polynomial covariants common for the whole isomorphic group family.

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