Ideal decompositions and computation of tensor normal forms

Abstract. 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.

• Type: Journal
• Year: 2002
• Where: CORR
• Authors: Bernd Fiedler
• Keywords: Certain Linear Subspaces, CORR 2002, Education, Symmetric Group Sr, Tensor