Geodesic
Ideal Decompositions and Computation of Tensor Normal Forms
Séminaire lotharingien de combinatoire, Tome 45 (2000-2001)

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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 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. 

@article{SLC_2000-2001_45_a6,
     author = {Bernd Fiedler},
     title = {Ideal {Decompositions} and {Computation} of {Tensor} {Normal} {Forms}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {45},
     year = {2000-2001},
     url = {http://geodesic.mathdoc.fr/item/SLC_2000-2001_45_a6/}
}
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Bernd Fiedler. Ideal Decompositions and Computation of Tensor Normal Forms. Séminaire lotharingien de combinatoire, Tome 45 (2000-2001). http://geodesic.mathdoc.fr/item/SLC_2000-2001_45_a6/