UNIVERSAL POINTS IN THE ASYMPTOTIC SPECTRUM OF TENSORS

• 成果类型: Article; Early Access
• 署名作者: Christandl, Matthias; Vrana, Peter; Zuiddam, Jeroen
• 署名单位: University of Copenhagen; Budapest University of Technology & Economics; University of Amsterdam
• 刊物名称: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
• ISSN/ISSBN: 0894-0347
• DOI: 10.1090/jams/996
• 发表日期: 2021
• 关键词: kronecker coefficients; matrix; complexity; sunflowers; polytopes; rank

摘要

1.1. The asymptotic restriction problem. We study the asymptotic restriction problem, following the pioneering work of Volker Strassen [57???60]. The asymptotic restriction problem is a problem about multilinear maps f : Fn1 x ?? ?? ?? x Fnk ??? F over an arbitrary field F. Letting (e1, ... , eni) be the standard basis of Fni, one may equivalently think of f as the k-tensor t E Fn1 ?? ?? ?? ?? ?? Fnk defined by t = ?? f (ea1, ..., eak)ea1 ?? ?????? ?? eak where ai ranges over {1,. .., nil. To state the asymptotic restriction problem we need the concepts of restriction and tensor product. Let f : Fn1 x ?? ?? ?? x Fnk ??? F and g : Fm1 x ?? ?? ?? x Fmk ??? F be multilinear maps. We say f restricts to g, and write f > g, if there are linear maps Ai : Fmi ??? Fni such that g = f o (A1, ... , Ak) where o denotes composition. We naturally define the tensor product f ??g as the multilinear map (Fn1 ??Fm1) x ?? ?? ?? x (Fnk ??Fmk) ??? F defined by (v1 ?? w1, ... , vk ?? wk) ?????? f (v1, ... , vk)g(w1, ... , wk). We say f restricts asymptotically to g, written f > g, if there is a sequence of natural numbers a(n) E