A Conjecture about Conserved Symmetric Tensors

TL;DR

This paper explores a conjecture about symmetric, conserved tensors, proposing that their integral moments vanish under certain conditions, supported by specific case proofs and related matrix results.

Contribution

It introduces a conjecture on conserved symmetric tensors' integral moments and provides proofs for particular cases, advancing understanding in tensor analysis.

Findings

Integral moments vanish when the number of coordinates is less than the tensor's rank in specific cases.

Derived results for large matrices generated by permutations.

Partial proofs supporting the conjecture under certain dimensionality assumptions.

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.