2017/07/23 by Michel Lavrauw, Lavrauw, Michel, Tomasz Popiel +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1707.07323
openalex publication_date 2017/07/23 · openalex created_date 2017/07/31 · openalex updated_date 2026/07/28
Let \mathbbF be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space V=\mathbbF3 ⊗ \mathbbF3 of 3 × 3 matrices over \mathbbF, and let G ≤ PGL(V) be the setwise stabiliser of the corresponding Segre variety S3,3(\mathbbF) in the projective space PG(V). The G-orbits of lines in PG(V) were determined by the first author and Sheekey as part of their classification of tensors in \mathbbF2 ⊗ V in the article "Canonical forms of 2 × 3 × 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 \mathbbF-span of the Veronese variety V3(\mathbbF) ⊂ S3,3(\mathbbF) under the natural action of K=PGL(3,\mathbbF). 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 \mathbbF. The corresponding orbit sizes and stabiliser subgroups of K are also determined in the case where \mathbbF is a finite field, and connections are drawn with old work of Jordan, Dickson and Campbell on the classification of pencils of conics in PG(2,\mathbbF), or equivalently, of pairs of ternary quadratic forms over \mathbbF.