[Paper Review] Canonical forms of $2 imes 3 imes 3$ tensors over the real field, algebraically closed fields, and finite fields

Michel Lavrauw, John Sheekey|arXiv (Cornell University)|Nov 21, 2014

Finite Group Theory Research5 references3 citations

TL;DR

This paper classifies the orbits of $2\times3\times3$ tensors over finite fields, the real numbers, and algebraically closed fields under the actions of $G = \mathrm{GL}(2)\times\mathrm{GL}(3)\times\mathrm{GL}(3)$ and $H = \mathrm{GL}(2)\times\mathrm{GL}(3)\times\mathrm{GL}(3)$, using geometric methods based on the Segre embedding. The key result is a complete enumeration: 21 $H$-orbits and 18 $G$-orbits over finite fields, 18 $H$-orbits and 15 $G$-orbits over algebraically closed fields, and 20 $H$-orbits and 17 $G$-orbits over the reals, with explicit canonical forms provided for each.

ABSTRACT

We classify the orbits of elements of the tensor product spaces ${\mathbb{F}}^2\otimes {\mathbb{F}}^3 \otimes {\mathbb{F}}^3$ for all finite; real; and algebraically closed fields under the action of two natural groups. The result can also be interpreted as the classification of the orbits in the $17$-dimensional projective space of the Segre variety product of a projective line and two projective planes. This extends the classification of the orbits in the $7$-dimensional projective space of the Segre variety product of three projective lines [M. Lavrauw and J. Sheekey: Orbits of the stabiliser group of the Segre variety product of three projective lines, Finite Fields Appl. (2014)]. The proof is geometric in nature, relies on properties of the Segre embedding, and uses the terminology of projective spaces.

Motivation & Objective

• To classify the orbits of $2\times3\times3$ tensors in $\mathbb{F}^2\otimes\mathbb{F}^3\otimes\mathbb{F}^3$ under the action of $G$ and $H$ for finite fields, the real field, and algebraically closed fields.
• To provide geometric characterizations of each orbit using the Segre embedding and projective geometry.
• To extend previous classifications—particularly the $2\times2\times2$ case—to the $2\times3\times3$ case, offering a unified, field-independent approach.
• To recover and re-derive known results over $\mathbb{C}$ and $\mathbb{R}$, and to correct or clarify inconsistencies in prior work, especially over finite fields.
• To lay the foundation for classifying orbits in higher-dimensional spaces such as $\mathbb{F}^2\otimes\mathbb{F}^3\otimes\mathbb{F}^r$.

Proposed method

• The authors use geometric techniques based on the Segre embedding $\sigma_{2,3,3}: \mathrm{PG}(1)\times\mathrm{PG}(2)\times\mathrm{PG}(2) \to \mathrm{PG}(17)$, mapping the product of projective spaces to the projective space of the tensor product.
• They analyze the orbits of tensors under the action of $G$ and $H$, where $G$ is the stabilizer of fundamental tensors and $H = \mathrm{GL}(2)\times\mathrm{GL}(3)\times\mathrm{GL}(3)$.
• The classification is achieved by studying the rank distribution $r_1(A)$ of the first contraction space of a representative tensor $A$, along with invariants such as the number of linearly independent rank-one components.
• The proof relies on field-independent geometric reasoning, avoiding topological or algebraic methods that fail over finite fields, and uses canonical forms derived from symmetric and skew-symmetric bilinear forms.
• For finite fields, the classification is completed by analyzing the non-vanishing of certain quadratic and cubic polynomials over $\mathbb{F}$, such as $v\lambda^2 + uv\lambda - 1 \neq 0$ for all $\lambda \in \mathbb{F}$.
• The authors derive explicit canonical forms for each orbit, using a fixed basis $e_1,e_2,e_3$ of $\mathbb{F}^3$ and the tensor $e = e_1\otimes e_1 + e_2\otimes e_2 + e_3\otimes e_3$.

Experimental results

Research questions

• RQ1 How many $H$-orbits exist in $\mathbb{F}^2\otimes\mathbb{F}^3\otimes\mathbb{F}^3$ when $\mathbb{F}$ is a finite field, and what are their canonical forms?
• RQ2 What is the number of $G$-orbits in $\mathbb{F}^2\otimes\mathbb{F}^3\otimes\mathbb{F}^3$ over algebraically closed fields, and how do they differ from the $H$-orbits?
• RQ3 How does the classification of $2\times3\times3$ tensor orbits over $\mathbb{R}$ compare to that over finite and algebraically closed fields?
• RQ4 Can the canonical forms for $2\times3\times3$ tensors over finite fields be derived without relying on complex or topological methods?
• RQ5 What is the relationship between the $G$-orbits in $\mathbb{F}^2\otimes\mathbb{F}^3\otimes\mathbb{F}^3$ and the $G$-orbits in the $2\times2\times3$ subcase?

Key findings

• Over finite fields, there are exactly 21 $H$-orbits and 18 $G$-orbits in $\mathbb{F}^2\otimes\mathbb{F}^3\otimes\mathbb{F}^3$, with canonical forms explicitly listed for each.
• Over algebraically closed fields, there are exactly 18 $H$-orbits and 15 $G$-orbits, confirming the classification extends to all algebraically closed fields.
• Over the real numbers, there are exactly 20 $H$-orbits and 17 $G$-orbits, reflecting the richer structure of real bilinear forms and signature invariants.
• The canonical forms are derived using rank distributions $r_1(A)$ and field-specific conditions such as the non-vanishing of quadratic or cubic polynomials over $\mathbb{F}$.
• The classification recovers and re-proves known results over $\mathbb{C}$ and $\mathbb{R}$, and corrects inconsistencies in prior work, particularly for even characteristic finite fields.
• The results allow a full classification of orbits in $\mathbb{F}^2\otimes\mathbb{F}^3\otimes\mathbb{F}^r$ for any $r$, as noted in the forthcoming work [11].