Rank and symmetries of signature tensors

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.