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Determination of the structure of algebraic curvature tensors by means of Young symmetrizers. (English) Zbl 1043.53016
Summary: 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.

53B20 Local Riemannian geometry
05E10 Combinatorial aspects of representation theory
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
05-04 Software, source code, etc. for problems pertaining to combinatorics
15A72 Vector and tensor algebra, theory of invariants
PERMS; Mathematica; Ricci
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