2021/11/14 by Xin Li, Liping Zhang, Li, Xin +3
Mathematics · Computer Science · #Tensor decomposition and applications #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2111.07343
Motivated by the problems raised by Bürgisser and Ikenmeyer, we discuss two classes of minimal generic fundamental invariants for tensors of order 3. The first one is defined on ⊗3 ℂm, where m=n2-1. We study its construction by obstruction design introduced by Bürgisser and Ikenmeyer, which partially answers one problem raised by them. The second one is defined on ℂℓ m⊗ ℂmn⊗ ℂnℓ. We study its evaluation on the matrix multiplication tensor ⟨ℓ,m,n⟩ and unit tensor ⟨ n2 ⟩ when ℓ=m=n. The evaluation on the unit tensor leads to the definition of Latin cube and 3-dimensional Alon-Tarsi problem. We generalize some results on Latin square to Latin cube, which enrich the understanding of 3-dimensional Alon-Tarsi problem. It is also natural to generalize the constructions to tensors of other orders. We illustrate the distinction between even and odd dimensional generalizations by concrete examples. Finally, some open problems in related fields are raised.