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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*[*S*_{r}] of a symmetric group
*S*_{r}.
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*[*S*_{r}]
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 *S*_{r} 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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