发表机构
Max Planck Institute for Mathematics in the Sciences; Technical University of Berlin（马克斯·普朗克科学促进学会数学与科学研究所; 柏林工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究将方阵可分解为对称与斜对称矩阵之和的事实推广到高阶张量，通过线性代数、表示论和组合学视角研究张量空间，其源于多领域，还表明张量空间分解可确定分段线性路径签名的消失理想。
AI 中文摘要
每个方阵都是一个对称矩阵和一个斜对称矩阵的唯一和。我们将这个熟悉的事实推广到高阶张量：对于每个\(m\leq k\)的选择，每个三次\(k\) - 张量都是一个\(m\) - 分段对称张量和一个\((k - m)\) - 分段斜对称张量的唯一和。我们从线性代数、表示理论和组合学的角度研究这些张量空间。我们的动机来自于随机分析、代数几何和数据科学中的签名张量。更具体地说，我们表明我们的张量空间分解确定了具有固定段数的分段线性路径签名的消失理想。
英文摘要
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 $m\leq k$. 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.
Comments26 pages, 1 figure
