Determination of the structure of algebraic curvature tensors by means of Young symmetrizers

2002/12/23 by B. Fiedler
Mathematics · Computer Science · #math.CO #cs.SC #math.DG #msc:53B20 #msc:15A72 #msc:05E10 #msc:16D60 #msc:05-04

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published as Seminaire Lotharingien de Combinatoire, 48 (2003) Article B48d · 19 pages. To appear Seminaire Lotharingien de Combinatoire: http://www.mat.univie.ac.at/~slc/

arxiv created 2002/12/23 · arxiv updated 2009/11/30

Abstract

For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins that every algebraic curvature tensor has a structure which is very similar to that of the above Osserman curvature tensors. We verify our results by means of the Littlewood-Richardson rule and plethysms. For certain symbolic calculations we used the Mathematica packages MathTensor, Ricci and PERMS.

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