Séminaire Lotharingien de Combinatoire, B45g (2001), 16 pp.

Bernd Fiedler

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 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.

Received: February 6, 2001; Accepted: April 5, 2001.

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