Two classes of minimal generic fundamental invariants for tensors

TL;DR

The paper advances the theory of minimal generic fundamental invariants for order-3 tensors by introducing two invariant families, F_m on $\\otimes^3 \\\mathbb{C}^{n^2-1}$ (via diagonal-obstruction designs) and F_{(\\ell,m,n)} on $\\mathbb{C}^{\\ell m}\\otimes \\mathbb{C}^{mn}\\otimes \\mathbb{C}^{n\\ell}$ (a noncubic generalization). It establishes nonvanishing results through Kronecker coefficients (notably when $k((n^2 ext{-}1)\\times n, (n^2 ext{-}1)\\times n, (n^2 ext{-}1)\\times n)=1$ and $k(\\ell m\\times n, mn\\times \\ell, n\\ell\\times m)=1$), and analyzes evaluations on canonical tensors such as the matrix multiplication tensor $\\langle\\ell,m,n\\rangle$ and the unit tensor $\\langle n^2\\rangle$. A deep link is drawn between these invariants and Latin cube combinatorics, via the 3D Alon-Tarsi problem, including concrete computations (e.g., $F_{(2,2,2)}(\\langle2,2,2\\rangle)=864$, $F_{(2,2,3)}(\\langle2,2,3\\rangle)=181{,}440$) and general constructions using obstruction designs. The work further extends to even/odd dimensional generalizations and higher-order tensors, discusses the corresponding obstruction-design classifications, and raises open problems on combinatorial and geometric aspects of these invariants.

Abstract

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 $\otimes^3 \mathbb{C}^m$, where $m=n^2-1$. 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 $\mathbb{C}^{\ell m}\otimes \mathbb{C}^{mn}\otimes \mathbb{C}^{n\ell}$. 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.

Figures (1)

• Figure 1: The obstruction design $B(3,3,3)$

Theorems & Definitions (55)

• Definition 2.1
• Theorem 3.1
• proof
• Remark 3.2
• Definition 3.3
• Lemma 3.4
• Definition 3.5
• Lemma 3.6
• Lemma 3.7
• proof