The symmetric representation of lines in $\text{PG}({\mathbb{F}}^3\otimes {\mathbb{F}}^3)$

Abstract

Let $\mathbb{F}$ be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space $V={\mathbb{F}}^3\otimes {\mathbb{F}}^3$ of $3\times 3$ matrices over $\mathbb{F}$, and let $G\le \text{PGL}(V)$ be the setwise stabiliser of the corresponding Segre variety $S_{3,3}(\mathbb{F})$ in the projective space $\text{PG}(V)$. The $G$-orbits of lines in $\text{PG}(V)$ were determined by the first author and Sheekey as part of their classification of tensors in ${\mathbb{F}}^2\otimes V$ in the article "Canonical forms of $2\times 3\times 3$ tensors over the real field, algebraically closed fields, and finite fields", Linear Algebra Appl. 476 (2015) 133-147. Here we consider the related problem of classifying those line orbits that may be represented by {\em symmetric} matrices, or equivalently, of classifying the line orbits in the $\mathbb{F}$-span of the Veronese variety ${\mathcal{V}}_3(\mathbb{F})\subset S_{3,3}(\mathbb{F})$ under the natural action of $K=\text{PGL}(3,\mathbb{F})$. Interestingly, several of the $G$-orbits that have symmetric representatives split under the action of $K$, and in many cases this splitting depends on the characteristic of $\mathbb{F}$. The corresponding orbit sizes and stabiliser subgroups of $K$ are also determined in the case where $\mathbb{F}$ is a finite field, and connections are drawn with old work of Jordan, Dickson and Campbell on the classification of pencils of conics in $\text{PG}(2,\mathbb{F})$, or equivalently, of pairs of ternary quadratic forms over $\mathbb{F}$.

References in corpus (2)

• Veronese Varieties over Fields with non-zero Characteristic: A Survey
• On embedded products of Grassmannians