Ideal decompositions and computation of tensor normal forms

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Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[Sᵣ] of a symmetric group Sᵣ. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W of W within the dual space of K[Sᵣ] yield linear identities needed for a treatment of the term combination problem for the coordinates of the T. We give the structure of these W for every situation which appears in symbolic tensor calculations by computer. Characterizing idempotents of such W can be determined by means of an ideal decomposition algorithm which works in every semisimple ring up to an isomorphism. Furthermore, we use tools such as the Littlewood-Richardson rule, plethysms and discrete Fourier transforms for Sᵣ to increase the efficience of calculations. All described methods were implemented in a Mathematica package called PERMS.