Piecewise Symmetric Tensors

Every square matrix is uniquely the sum of a symmetric matrix and a skew-symmetric matrix. We extend this familiar fact to higher order tensors: every cubic $k$-tensor is uniquely the sum of an $m$-piecewise symmetric tensor and a $(k\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}m)$-piecewise skew-symmetric tensor, for each choice of $m\le k$. We study these tensor spaces from the perspectives of linear algebra, representation theory and combinatorics. Our motivation stems from signature tensors in stochastic analysis, algebraic geometry and data science. More specifically, we show that our tensor space decompositions determine the vanishing ideals for signatures of piecewise linear paths with a fixed number of segments.