  The signature of a path is a sequence of tensors whose entries are iterated
integrals, playing a key role in stochastic analysis and applications. The set
of all signature tensors at a particular level gives rise to the universal
signature variety. We show that the parametrization of this variety induces a
natural decomposition of the tensor space via representation theory, and
connect this to the study of path invariants. We also examine the question of
determining what is the tensor rank of a signature tensor.
