Two classes of minimal generic fundamental invariants for tensors

TL;DR

This paper introduces two classes of minimal fundamental invariants for tensors, explores their properties, and connects them to Latin cubes and the 3D Alon-Tarsi problem, advancing understanding in tensor invariants and combinatorial structures.

Contribution

It presents two new classes of minimal generic fundamental invariants for order-3 tensors and links them to Latin cubes and the 3D Alon-Tarsi problem, addressing open questions.

Findings

Construction of invariants via obstruction design

Evaluation on matrix multiplication and unit tensors

Extension of Latin square results to Latin cubes

Abstract

Motivated by the problems raised by B\"{u}rgisser and Ikenmeyer, we discuss two classes of minimal generic fundamental invariants for tensors of order 3. The first one is defined on $\otimes^3 \mathbb{C}^m$, where $m=n^2-1$. We study its construction by obstruction design introduced by B\"{u}rgisser and Ikenmeyer, which partially answers one problem raised by them. The second one is defined on $\mathbb{C}^{\ell m}\otimes \mathbb{C}^{mn}\otimes \mathbb{C}^{n\ell}$. We study its evaluation on the matrix multiplication tensor $\langle\ell,m,n\rangle$ and unit tensor $\langle n^2 \rangle$ when $\ell=m=n$. 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.