A Conjecture about Conserved Symmetric Tensors

Abstract

We consider T(x), a tensor of arbitrary rank that is symmetric in all of its indices and conserved in the sense that the divergence on any one index vanishes. Our conjecture is that all integral moments of this tensor will vanish if the number of coordinates in that integral moment is less than the rank of the tensor. This result is proved explicitly for a number of particular 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 matrices generated by permutations.