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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.

MSC:
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
Software:
PERMS; Mathematica; Ricci
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