Decomposing tensor spaces via path signatures

sponsorship: The authors thank Bernd Sturmfels for suggesting the topic. This project started while the authors were attending the Thematic Research Programme Algebraic Geometry with Applications to Tensors and Secants (AGATES) for which they acknowledge University of Warsaw, Excellence Initiative - Research University and the Simons Foundation Award No. 663281 granted to the Institute of Mathematics of the Polish Academy of Sciences for the years 2021-2023.Amendola acknowledges support from DFG CRC/TRR 388 "Rough Analysis, Stochastic Dynamics and Related Fields", Project A04 and Project B01. Galuppi acknowledges support from the National Science Center, Poland, projects "Complex contact manifolds and geometry of secants", 2017/26/E/ST1/00231, and "Tensor rank and its applications to signature tensors of paths", 2023/51/D/ST1/02363. Santarsiero was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 445466444 and by the European Union under NextGenerationEU. PRIN 2022, Prot. 2022E2Z4AK. Seynnaeve is supported by Research foundation - Flanders (FWO) - Grant Number 1219723N. (Simons Foundation|663281, National Science Center, Poland|2017/26/E/ST1/00231, National Science Center, Poland|2023/51/D/ST1/02363, Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)|CRC/TRR 388, Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)|445466444, European Union under NextGenerationEU. PRIN 2022|2022E2Z4AK, Research foundation - Flanders (FWO)|1219723N)