Alfred-Rosch-Str. 13, 04249 Leipzig, Germany

September, 1999

We consider the problem to determine
normal forms of the coordinates of covariant tensors
T Î *T*_{r} V
of order r over a finite-dimensional **K**-vector space,
**K** = **R** , **C**.
A connection between such tensors and the group ring
**K**[*S*_{r}] can be established by
assigning a group ring element
T_{b} : = å_{p Î Sr}T(v_{p(1)} , ¼, v_{p(r)}) p Î **K**[*S*_{r}]
to every tensor T Î *T*_{r} V and every r-tuple
b = (v_{1} , ¼, v_{r}) Î V^{r} of vectors. Then each symmetry class
*T* Í *T*_{r} V of tensors can be
characterized by a linear
subspace W Í **K**[*S*_{r}] which is spanned by all T_{b} of
the T Î *T*. The elements of the orthogonal subspace
W^{^} Í **K**[*S*_{r}]^{*} of W within the dual space
**K**[*S*_{r}]^{*} yield the set of all linear
identities that are
fulfiled by the coordinates of all tensors T Î *T*. These identities
can be used to calculate linearly independent coordinates (i.e. normal forms)
of the T Î *T*.

If the T Î *T* are single tensors and dimV ³ r, then W is a
left ideal W = **K**[*S*_{r}]·e generated by an idempotent e.
In the case of tensor products
T_{1} ÄT_{2} Ä¼ÄT_{m} or T Ä¼ÄT
(m-times), W is a left ideal whose structure is described by a
Littlewood-Richardson product [a_{1}][a_{2}]¼[a_{m}]
or a plethysm [ a] Ä[m], respectively. We have also treated
the cases in which
dimV < r or the
T Î *T* contain aditional contractions of index pairs. In these
cases
characterizing linear subspaces W Í **K**[*S*_{r}] with a
structure
W = f ·**K**[*S*_{r}]·e or
W = å_{i = 1}^{k} a_{i} ·**K**[*S*_{r}] ·e come into play.
Here e , f Î **K**[*S*_{r}] are idempotents.

We have implemented a Mathematica package by which the characterizing
idempotents and bases of the spaces W and the identities from W^{^}
can be determined in all above cases. This package contains an ideal
decomposition algorithm and tools such as the Littlewood-Richardson rule,
plethysms and discrete Fourier transforms.

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