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Tetrads of lines spanning \(\mathrm{PG}(7,2)\). (English) Zbl 1281.51007

Summary: Our starting point is a very simple one, namely that of a set \(\mathcal{L} _{4}\) of four mutually skew lines in \(\mathrm{PG}(7,2)\). Under the natural action of the stabilizer group \(\mathcal{G}(\mathcal{L}_{4} )<\mathrm{GL}(8,2)\) the \(255\) points of \(\mathrm{PG}(7,2)\) 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 \(\mathrm{PG}(7,2)\). This ties in with the recent finding that each \(\mathcal{S}=\mathcal{S}_{3}(2)\) in \(\mathrm{PG} (7,2)\) determines a distinguished \(Z_{3}\) subgroup of \(\mathrm{GL}(8,2)\) which generates two sibling copies \(\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\) of \(\mathcal{S}\).

MSC:

51E20 Combinatorial structures in finite projective spaces
05B25 Combinatorial aspects of finite geometries
15A69 Multilinear algebra, tensor calculus
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Full Text: arXiv Euclid