  The signature of a path is a sequence of tensors which allows to uniquely
reconstruct the path. In this paper we propose a systematic study of basic
properties of signature tensors, starting from their rank, symmetries and
conciseness. We prove a sharp upper bound on the rank of signature tensors of
piecewise linear paths. We show that there are no skew-symmetric signature
tensors of order three or more, and we also prove that specific instances of
partial symmetry can only happen for tensors of order three. Finally, we give a
simple geometric characterization of paths whose signature tensors are not
concise.
