Partial Degeneration of Tensors

• DOI: 10.48550/arxiv.2212.14095
• Publication Date: 2022-01-01
• Authors: Matthias Christandl; Fulvio Gesmundo; Vladimir Lysikov; Vincent Steffan
• Keywords: Rank (graph theory); Matrix (chemical analysis); Cartesian tensor

Tensors are often studied by introducing preorders such as restriction and degeneration: the former describes transformations of tensors local linear maps on its tensor factors; latter where may vary along a curve, resulting is expressed limit this curve. In work we introduce study partial degeneration, special version degeneration one constant whereas others Motivated algebraic complexity, quantum entanglement networks, present constructions based matrix multiplication find examples making connection to theory prehomogeneous spaces. We highlight subtleties new notion showing obstruction classification results for unit tensor. To end, aided rank, natural generalization rank. The existence degenerations gives strong upper bounds rank tensor, which allows turn into restrictions. particular, several examples, W-tensor Coppersmith-Winograd tensors, lower provide obstructions certain degenerations.