The slice rank of a direct sum
Combinatorics
2021-08-12 v2
Abstract
We show that the slice rank of the direct sum of two tensors is equal to the sum of their slice ranks. The upper bound is trivial, but the lower bound needs more than a one-line proof, for reasons we explain. This result generalizes the fact, shown by Tao, that the slice rank of a diagonal tensor is equal to the number of non-zero entries of that tensor.
Keywords
Cite
@article{arxiv.2105.08394,
title = {The slice rank of a direct sum},
author = {W. T. Gowers},
journal= {arXiv preprint arXiv:2105.08394},
year = {2021}
}
Comments
This corrects an earlier version, with a proof that proceeds along much more standard lines. The result is therefore less interesting than it might have seemed, but worth recording at least in preprint form in case it is ever needed