Abstract
The 155 points of the Grassmannian \(G_{1,4,2}\) of lines of PG (4, 2) = \(\mathbb{P}V\left( {5,2} \right)\) are those points \(x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)\) which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X \( \subset \) PG (9, 2) will be termed odd or even according as X intersects \(G_{1,4,2}\) in an odd or even number of points. Let \(Q^\ddag \left( {x_1 ,...,x_5 } \right)\) denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X # of a r-flat X \( \subset \) PG (9, 2) by
. Because \(Q^\ddag\) is quinquelinear, the associate X # of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X \( \subseteq\) X # while if X is an even 4-flat then X # is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X # = X. An example of an even 4-flat X such that \(\left( {X^\# } \right)^\#\) = X is provided by any 4-flat X which is external to \(G_{1,4,2}\). However, it appears that the two possibilities just illustrated, namely X # = X for an odd 4-flat and \(\left( {X^\# } \right)^\#\) = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X # = PG (9, 2) and of even 4-flats for which \({X^{\# \# \# } }\) = X.
Similar content being viewed by others
References
J. W. P. Hirschfeld and J. A. Thas, General Galois Geometries, Clarendon, Oxford (1991).
N. A. Gordon, R. Shaw and L. H. Soicher, Classification of partial spreads in PG(4; 2), (2000) 65 pp., accessible from: http://www.hull.ac.uk/maths/people/rs/staffdetails.html.
R. Shaw and N. A. Gordon, The lines of PG(4; 2) are the points of a quintic in PG(9; 2), J. Combin. Theory (A), Vol. 68 (1994) pp. 226–231.
R. Shaw and N. A. Gordon, The associate X # of a flat X in PG(n; 2) with respect to a given hypersurface, (2003) 60 pp., accessible from: http://www.hull.ac.uk/maths/people/rs/staffdetails.html.
R. Shaw, N. A. Gordon and J. G. Maks, Partial spreads in PG(4; 2) and flats in PG(9; 2) external to the Grassmannian \(G_{1,4,2}\). (Submitted to Proceedings of Combinatorics, 2002.)
R. Shaw, J. G. Maks and N. A. Gordon, The classification of flats in PG(9; 2) which are external to the Grassmannian \(G_{1,4,2}\). To appear in Designs, Codes and Cryptography.
R. Shaw and J. G. Maks, Conclaves of planes in PG(4; 2) and certain planes external to the Grassmannian \(G_{1,4,2}\) ⊂ PG(9; 2), J. Geom., Vol. 78 (2003) pp. 168–180. (Also accessible from: http://www.hull.ac.uk/maths/people/rs/staffdetails.html.)
R. Shaw, Linear Algebra and Group Representations, Vol. 2, Academic Press, London (1983).
R. Shaw, A characterization of the primals in PG(m; 2), Designs, Codes and Cryptography, Vol. 2 (1992) pp. 253–256.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shaw, R., Gordon, N.A. The Quintic Grassmannian \(G_{1,4,2}\) in PG(9, 2). Designs, Codes and Cryptography 32, 381–396 (2004). https://doi.org/10.1023/B:DESI.0000029236.10701.61
Issue Date:
DOI: https://doi.org/10.1023/B:DESI.0000029236.10701.61