Ideal decompositions and computation of tensor normal forms

2002/11/09 by B. Fiedler
Mathematics · Computer Science · #math.CO #cs.SC #math.DG #msc:16D60 #msc:15A72 #msc:05E10 #msc:16D70 #msc:16S50 #msc:05-04

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published as Seminaire Lotharingien de Combinatoire, 45 (2001) Article B45g. http://www.mat.univie.ac.at/~slc/wpapers/s45fiedler.html · 16 pages

arxiv created 2002/11/09 · arxiv updated 2009/11/30

Abstract

Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[S_r] of a symmetric group S_r. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W^\bot of W within the dual space of K[S_r] 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 S_r to increase the efficience of calculations. All described methods were implemented in a Mathematica package called PERMS.

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