The symmetric representation of lines in \(\operatorname{PG} ( \mathbb{F}^3 \otimes \mathbb{F}^3 )\). (English) Zbl 1433.05063

Summary: 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 \(G \leq \text{PGL} ( V )\) be the setwise stabiliser of the corresponding Segre variety \(S_{3 , 3} ( \mathbb{F} )\) in the projective space \(\operatorname{PG} ( V )\). The \(G\)-orbits of lines in \(\text{PG} ( V )\) were determined by M. Lavrauw and J. Sheekey [J. Geom. 108, No. 1, 5–23 (2017; Zbl 1361.05022)] as part of their classification of tensors in \(\mathbb{F}^2 \otimes V\). 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 \(\mathbb{F} \)-span of the Veronese variety \(\mathcal{V}_3 ( \mathbb{F} ) \subset S_{3 , 3} ( \mathbb{F} )\) under the natural action of \(K = \operatorname{PGL} ( 3 , \mathbb{F} )\). 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 \(\mathbb{F} \). Although our main focus is on the case where \(\mathbb{F}\) is a finite field, our methods (which are mostly geometric) are easily adapted to include the case where \(\mathbb{F}\) 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 \(\mathbb{F}\) is a finite field, and connections are drawn with old work of Jordan and Dickson on the classification of pencils of conics in \(\operatorname{PG} ( 2 , \mathbb{F} )\), or equivalently, of pairs of ternary quadratic forms over \(\mathbb{F} \).


05B25 Combinatorial aspects of finite geometries
51E20 Combinatorial structures in finite projective spaces
14N05 Projective techniques in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds


Zbl 1361.05022
Full Text: DOI arXiv


[1] Artamkin, D. I.; Nurmiev, A. G., Orbits and invariants of third-order cubic matrices with symmetric fibers, Mat. Zametki, 72, 4, 483-489 (2002), (in Russian); Translation in Math. Notes 72 (3-4) (2002) 447-453 · Zbl 1099.13011
[2] Bertini, E., Introduzione Alla Geometria Proiettiva Degli Iperspazi (1923), Principato: Principato Messina
[3] Campbell, A. D., Pencils of conics in the galois fields of order \(2^n\), Amer. J. Math., 49, 401-406 (1927)
[4] Campbell, A. D., Nets of conics in the galois field of order \(2^n\), Bull. Amer. Math. Soc., 34, 4, 481-489 (1928)
[5] Dickson, L. E., Linear Groups: With an Exposition of the Galois Field Theory (1901), B. G. Teubner: B. G. Teubner Leipzig
[6] Dickson, L. E., On families of quadratic forms in a general field, Quart. J. Pure Appl. Math., 45, 316-333 (1908)
[7] Gantmacher, F. R., The Theory of Matrices (1959), Chelsea: Chelsea Providence, RI · Zbl 0085.01001
[8] Harris, J., Algebraic Geometry: A First Course (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0779.14001
[9] Havlicek, H., Veronese varieties over fields with non-zero characteristic: a survey. Combinatorics 2000 (Gaeta), Discrete Math., 267, 1-3, 159-173 (2003) · Zbl 1030.51017
[10] Hirschfeld, J. W.P., Projective Geometries over Finite Fields (1998), Oxford University Press: Oxford University Press Oxford · Zbl 0899.51002
[11] Hirschfeld, J. W.P.; Thas, J. A., General Galois Geometries (2016), Springer-Verlag: Springer-Verlag London · Zbl 1358.51002
[12] Hodge, W. V.D.; Pedoe, D., Methods of Algebraic Geometry, vol. II (1952), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0048.14502
[13] Jordan, C., Réduction d’un réseau de formes quadratiques ou bilinéaires: première partie, J. Math. Pures Appl., 403-438 (1906)
[14] Jordan, C., Réduction d’un réseau de formes quadratiques ou bilinéaires: deuxième partie, J. Math. Pures Appl., 5-51 (1907)
[15] Lavrauw, M.; Sheekey, J., Canonical forms of \(2 \times 3 \times 3\) tensors over the real field, algebraically closed fields, and finite fields, Linear Algebra Appl., 476, 133-147 (2015) · Zbl 1314.05029
[16] Lavrauw, M.; Sheekey, J., Classification of subspaces in \(\mathbb{F}^2 \otimes \mathbb{F}^3\) and orbits in \(\mathbb{F}^2 \otimes \mathbb{F}^3 \otimes \mathbb{F}^r\), J. Geom., 108, 5-23 (2017) · Zbl 1361.05022
[17] Mazzocca, F.; Melone, N., Caps and Veronese varieties in projective Galois spaces, Discrete Math., 48, 2-3, 243-252 (1984) · Zbl 0537.51014
[18] Wall, C. T.C., Nets of conics, Math. Proc. Cambridge Philos. Soc., 81, 351-364 (1977) · Zbl 0351.14032
[19] Wilson, A. H., The canonical types of nets of modular conics, Amer. J. Math., 36, 2, 187-210 (1914)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.