Tetrads of lines spanning PG(7,2)

Shaw, Ron; Gordon, Neil; Havlicek, Hans; Shaw, Ronald

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.

• Publication Date: Jan 1, 2013
• Journal: Bulletin of the Belgian Mathematical Society - Simon Stevin
• Print ISSN: 1370-1444
• Peer Reviewed: Peer Reviewed
• Volume: 20
• Issue: 4
• Pages: 735-752
• Institution Citation: Shaw, R., Gordon, N., Havlicek, H., & Shaw, R. (2013). Tetrads of lines spanning PG(7,2). Bulletin of the Belgian Mathematical Society, Simon Stevin, 20(4), 735-752
• Keywords: Segre variety S₃(2), Line-spread, Invariant polynomials
• 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