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On Symmetric Irreducible Tensors In D-dimensions

Harald Ehrentraut, Wolfgang Muschik
Published 1998 · Mathematics
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This paper establishes the notion of symmetric irreducible tensors with arbitrary dimensions d ≥ 2. These tensors are generalisations of a symmetric traceless, second order tensor and their significance stems from their close connection to spherical harmonics. We introduce the general concepts and derive some fundamental relations with respect to these tensors. Special considerations are given to proofs, because those are hard to find, or absent, in the available literature. The relation to spherical harmonics mentioned above is used to obtain a theorem which allows L2(Sd)-functions to be expressed as a tensor series with respect to symmetric irreducible tensors. Such representations can be (and have been) used in the theory of liquid crystals to deal with orientation distribution functions and to introduce suitable order parameters.

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