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october 2013 Tetrads of lines spanning $\operatorname*{PG}(7,2)$
Ron Shaw, Neil Gordon, Hans Havlicek
Bull. Belg. Math. Soc. Simon Stevin 20(4): 735-752 (october 2013). DOI: 10.36045/bbms/1382448192

Abstract

Our starting point is a very simple one, namely that of a set $\mathcal{L} _{4}$ of four mutually skew lines in $\operatorname*{PG}(7,2).$ Under the natural action of the stabilizer group $\mathcal{G}(\mathcal{L}_{4} )<\operatorname*{GL}(8,2)$ the $255$ points of $\operatorname*{PG}(7,2)$ fall into four orbits $\omega_{1},\omega_{2},\omega_{3},\omega_{4},$ of respective lengths $12,54,108,81.$ We show that the $135$ points $\in\omega_{2}\cup \omega_{4}$ are the internal points of a hyperbolic quadric $\mathcal{H}_{7}$ determined by $\mathcal{L}_{4},$ and that the $81$-set $\omega_{4}$ (which is shown to have a sextic equation) is an orbit of a normal subgroup $\mathcal{G} _{81}\cong(Z_{3})^{4}$ of $\mathcal{G}(\mathcal{L}_{4}).$ There are $40$ subgroups $\cong(Z_{3})^{3}$ of $\mathcal{G}_{81},$ and each such subgroup $H<\mathcal{G}_{81}$ gives rise to a decomposition of $\omega_{4}$ into a triplet $\{\mathcal{R}_{H},\mathcal{R}_{H}^{\prime},\mathcal{R}_{H} ^{\prime\prime}\}$ of $27$-sets. We show in particular that the constituents of precisely $8$ of these $40$ triplets are Segre varieties $\mathcal{S} _{3}(2)$ in $\operatorname*{PG}(7,2).$ This ties in with the recent finding 225-239 --- that each $\mathcal{S}=\mathcal{S}_{3}(2)$ in $\operatorname*{PG} (7,2)$ determines a distinguished $Z_{3}$ subgroup of $\operatorname*{GL} (8,2)$ which generates two sibling copies $\mathcal{S}^{\prime},\mathcal{S} ^{\prime\prime}$ of $\mathcal{S}.$

Citation

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Ron Shaw. Neil Gordon. Hans Havlicek. "Tetrads of lines spanning $\operatorname*{PG}(7,2)$." Bull. Belg. Math. Soc. Simon Stevin 20 (4) 735 - 752, october 2013. https://doi.org/10.36045/bbms/1382448192

Information

Published: october 2013
First available in Project Euclid: 22 October 2013

zbMATH: 1281.51007
MathSciNet: MR3129071
Digital Object Identifier: 10.36045/bbms/1382448192

Subjects:
Primary: 05B25 , 15A69 , 51E20

Keywords: $\mathcal{S}_{3}(2)$ , invariant polynomials , line-spread , Segre variety

Rights: Copyright © 2013 The Belgian Mathematical Society

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Vol.20 • No. 4 • october 2013
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