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

Bernd Fiedler ()

A chapter in Algebraic Combinatorics and Applications, 2001, pp 118-133 from Springer

Abstract: 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 ⋅ K [ S r ] ⋅ e o r W = ∑ i = 1 k a i ⋅ K [ S r ] ⋅ e $$W = f \cdot \mathbb{K}[{{S}_{r}}] \cdot e\,or\,W = \sum\nolimits_{{i = 1}}^{k} {{{a}_{i}} \cdot \mathbb{K}[{{S}_{r}}] \cdot e}$$ 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.

Keywords: Normal Form; Discrete Fourier Transform; Linear Subspace; Left Ideal; Symmetry Class (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-59448-9_9

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DOI: 10.1007/978-3-642-59448-9_9

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