Characterization of Tensor Symmetries by Group Ring Subspaces and Computation of Normal Forms of Tensor Coordinates

Zusammenfassung

We consider the problem to determine normal forms of the coordinates of covariant tensors T ∈ TrV 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 MATHYPE to every tensor T ∈ TrV and every r-tuple b = (v1,…,vr) ∈ Vr of vectors. Then each symmetry class T ⊆ TrV 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 fulfilled 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 dim V ≥ 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 LittlewoodRichardson product [a1][a2] … [am] or a plethysm [a] 2299 [m], respectively. We have also treated the cases in which dim V 003C r or the T 2208 T contain aditional contractions of index pairs. In these cases characterizing linear subspaces W 2286 𝕂 [Sr] with a structure $W = f \cdot \mathbb{K}[{{S}_{r}}] \cdot e\,or\,W = \sum\nolimits_{{i = 1}}^{k} {{{a}_{i}} \cdot \mathbb{K}[{{S}_{r}}] \cdot e}$ 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 ⊥ 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.