Ron Shaw
Tetrads of lines spanning PG(7,2)
Shaw, Ron; Gordon, Neil; Havlicek, Hans
Abstract
Our starting point is a very simple one, namely that of a set L₄ of four mutually skew lines in PG(7,2). Under the natural action of the stabilizer group G(L₄)<GL(8,2) the 255 points of PG(7,2) fall into four orbits ω₁,ω₂,ω₃,ω4, of respective lengths 12,54,108,81. We show that the 135 points ∈ω₂∪ω₄ are the internal points of a hyperbolic quadric H7 determined by L₄, and that the 81-set ω₄ (which is shown to have a sextic equation) is an orbit of a normal subgroup G₈₁≅(Z₃)4 of G(L4). There are 40 subgroups ≅(Z₃)3 of G₈₁, and each such subgroup H<G₈₁ gives rise to a decomposition of ω4 into a triplet {RH,R′H,R′′H} of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S₃(2) in PG(7,2). This ties in with the recent finding 225-239 --- that each S=S₃(2) in PG(7,2) determines a distinguished Z₃ subgroup of GL(8,2) which generates two sibling copies S′,S′′ of S.
Citation
Shaw, R., Gordon, N., & Havlicek, H. (2013). Tetrads of lines spanning PG(7,2). Bulletin of the Belgian Mathematical Society, Simon Stevin, 20(4), 735-752. https://doi.org/10.36045/bbms/1382448192
Journal Article Type | Article |
---|---|
Acceptance Date | Feb 1, 2012 |
Publication Date | Jan 1, 2013 |
Deposit Date | Sep 30, 2015 |
Publicly Available Date | Nov 23, 2017 |
Journal | Bulletin of the Belgian Mathematical Society - Simon Stevin |
Print ISSN | 1370-1444 |
Peer Reviewed | Peer Reviewed |
Volume | 20 |
Issue | 4 |
Pages | 735-752 |
DOI | https://doi.org/10.36045/bbms/1382448192 |
Keywords | Segre variety S₃(2), Line-spread, Invariant polynomials |
Public URL | https://hull-repository.worktribe.com/output/379399 |
Publisher URL | https://projecteuclid.org/euclid.bbms/1382448192#abstract |
Additional Information | Authors' accepted manuscript of article published in: Bulletin of the Belgian Mathematical Society - Simon Stevin, 2013, v.20, issue 4 |
Contract Date | Nov 23, 2017 |
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Copyright Statement
©2015 University of Hull
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