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