Séminaire Lotharingien de Combinatoire, B48d (2003), 20 pp.
Determination of the Structure of Algebraic Curvature Tensors by
Means of Young Symmetrizers
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.
Received: June 30, 2001.
Accepted: October 5, 2001.
Final version: December 31, 2002.
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