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A Conjecture about Conserved Symmetric Tensors


We consider T(x), a tensor of arbitrary rank that is symmetric in all of its indicesand conserved in the sense that the divergence on any one index vanishes. Our conjecture isthat all integral moments of this tensor will vanish if the number of coordinates in that integralmoment is less than the rank of the tensor. This result is proved explicitly for a number ofparticular cases, assuming adequate dimensionality of the Euclidean space of coordinates (x);but a general proof is lacking. Along the way, we find some neat results for certain large matricesgenerated by permutations.

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