Lavrauw, Michel and Popiel, Tomasz (2020) The symmetric representation of lines in PG(F3 circle times F3). Discrete Mathematics, 343 (4). ISSN 0012365X (Print) 1872681X (Online)
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Official URL: http://dx.doi.org/10.1016/j.disc.2019.111775
Abstract
Let F be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space V = F3 circle times F3 of 3 x 3 matrices over F, and let G <= PGL(V) be the setwise stabiliser of the corresponding Segre variety S3,S3(F) in the projective space PG(V). The Gorbits of lines in PG(V) were determined by the first author and Sheekey as part of their classification of tensors in F2 circle times V in [15]. Here we solve the related problem of classifying those line orbits that may be represented by symmetric matrices, or equivalently, of classifying the line orbits in the Fspan of the Veronese variety V3(F) subset of S3,S3(F) under the natural action of K = PGL(3, F). Interestingly, several of the Gorbits that have symmetric representatives split under the action of K, and in many cases this splitting depends on the characteristic of F. Although our main focus is on the case where F is a finite field, our methods (which are mostly geometric) are easily adapted to include the case where F is an algebraically closed field, or the field of real numbers. The corresponding orbit sizes and stabiliser subgroups of K are also determined in the case where IF is a finite field, and connections are drawn with old work of Jordan and Dickson on the classification of pencils of conics in PG(2, F), or equivalently, of pairs of ternary quadratic forms over F.
Item Type:  Article 

Uncontrolled Keywords:  Veronese variety; Segre variety; Tensor product; Pencils of conics; Quadratic form 
Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences 
Depositing User:  Michel Lavrauw 
Date Deposited:  14 Sep 2020 12:54 
Last Modified:  29 Jul 2023 11:56 
URI:  https://research.sabanciuniv.edu/id/eprint/40073 
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The symmetric representation of lines in PG(. (deposited 29 Jul 2019 23:10)
 The symmetric representation of lines in PG(F3 circle times F3). (deposited 14 Sep 2020 12:54) [Currently Displayed]