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

Bernd Fiedler
Organization: Mathematical Institute, University of Leipzig
Journal / Anthology

Seminaire Lotharingien de Combinatoire
Year: 2003
Volume: 48
Page range: article B48d

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.

*Mathematics > Algebra > Linear Algebra