Canonical forms of Order-$k$ ($k = 2, 3, 4$) Symmetric Tensors of Format $3 \times \dots \times 3$ Over Prime Fields

TL;DR

This paper classifies symmetric tensors of small format over prime fields by their equivalence classes, ranks, and orbits under group actions, providing a comprehensive computational analysis of their canonical forms.

Contribution

It computes the canonical forms, ranks, and orbits of symmetric tensors of specific small formats over prime fields, a novel classification for these tensor types.

Findings

Classified tensors into equivalence classes under $GL_3(F_p)$

Determined maximum symmetric ranks for each tensor type

Compared symmetric rank with maximum rank for these tensors

Abstract

We consider symmetric tensors of format: $3 \times 3$ over $\mathbb{F}_p$ for $p = 2, 3, 5$; $3 \times 3 \times 3$ over $\mathbb{F}_p$ for $p = 2, 3$; and $3 \times 3 \times 3 \times 3$ over $\mathbb{F}_p$ for $p = 2, 3$. In each case we compute their equivalence classes under the action of the general linear group $GL_3 (\mathbb{F}_p )$. We use computer algebra to determine the set of tensors of each symmetric rank, then we compute the orbit of the group action. We determine the maximum symmetric rank of these tensors and compare it with the maximum rank.