Characterization of tensor symmetries by group ring subspaces and computation of normal forms of tensor coordinates

Bernd Fiedler
Alfred-Rosch-Str. 13, 04249 Leipzig, Germany
September, 1999


We consider the problem to determine normal forms of the coordinates of covariant tensors T Tr V of order r over a finite-dimensional K-vector space, K = R , C. A connection between such tensors and the group ring K[Sr] can be established by assigning a group ring element Tb : = p SrT(vp(1) , , vp(r)) p K[Sr] to every tensor T Tr V and every r-tuple b = (v1 , , vr) Vr of vectors. Then each symmetry class T Tr V of tensors can be characterized by a linear subspace W K[Sr] which is spanned by all Tb of the T T. The elements of the orthogonal subspace W^ K[Sr]* of W within the dual space K[Sr]* yield the set of all linear identities that are fulfiled by the coordinates of all tensors T T. These identities can be used to calculate linearly independent coordinates (i.e. normal forms) of the T T.

If the T T are single tensors and dimV r, then W is a left ideal W = K[Sr]·e generated by an idempotent e. In the case of tensor products T1 T2 Tm or T T (m-times), W is a left ideal whose structure is described by a Littlewood-Richardson product [a1][a2][am] or a plethysm [ a] [m], respectively. We have also treated the cases in which dimV < r or the T T contain aditional contractions of index pairs. In these cases characterizing linear subspaces W K[Sr] with a structure W = f ·K[Sr]·e or W = i = 1k ai ·K[Sr] ·e come into play. Here e , f K[Sr] 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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