Ideal decompositions and computation of tensor normal forms

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 K[Sr] 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. 1. The Term Combination Problem for Tensors The use of computer algebra systems for symbolic calculations with tensor expressions is very important in differential geometry, tensor analysis and general relativity theory. The investigations of this paper are motivated by the following term combination problem or normal form problem which occurs within such calculations. Let us consider real or complex linear combinations