Piecewise Symmetric Tensors
Abstract
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−m)-piecewise skew-symmetric tensor, for each choice of m≤ k. We study these tensor spaces via the combinatorics of word descents and compositions. Under Schur--Weyl duality, the two spaces naturally decompose into descent representations of the symmetric group. In fact, we prove a variant of Solomon's classical decomposition of its group algebra that explains this decomposition. 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 of the signatures of piecewise linear paths with a fixed number of segments.
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BibTeX
@misc{lotter2026piecewise,
title = {{Piecewise Symmetric Tensors}},
author = {Felix Lotter and Rosa Preiß},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04712},
eprint = {2607.04712},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04712},
}
RIS
TY - UNPB
TI - Piecewise Symmetric Tensors
AU - Felix Lotter
AU - Rosa Preiß
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2607.04712
UR - https://arxiv.org/abs/2607.04712
AB - 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−m)-piecewise skew-symmetric tensor, for each choice of m≤ k. We study these tensor spaces via the combinatorics of word descents and compositions. Under Schur--Weyl duality, the two spaces naturally decompose into descent representations of the symmetric group. In fact, we prove a variant of Solomon's classical decomposition of its group algebra that explains this decomposition. 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 of the signatures of piecewise linear paths with a fixed number of segments.
ER -