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Tetrads of lines spanning PG(7,2)

Ron Shaw, Neil Gordon, Hans Havlicek

Abstract

Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of 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 H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S.

Tetrads of lines spanning PG(7,2)

Abstract

Our starting point is a very simple one, namely that of a set L_4 of four mutually skew lines in PG(7,2): Under the natural action of the stabilizer group G(L_4) < GL(8,2) the 255 points of 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 H_7 determined by L_4; and that the 81-set omega_4 (which is shown to have a sextic equation) is an orbit of a normal subgroup G_81 isomorphic to (Z_3)^4 of G(L_4): There are 40 subgroups (isomorphic to (Z_3)^3) of G_81; and each such subgroup H < G_81 gives rise to a decomposition of omega_4 into a triplet of 27-sets. We show in particular that the constituents of precisely 8 of these 40 triplets are Segre varieties S_3(2) in PG(7,2): This ties in with the recent finding that each Segre S = S_3(2) in PG(7,2) determines a distinguished Z_3 subgroup of GL(8,2) which generates two sibling copies S'; S" of S.

Paper Structure

This paper contains 13 sections, 17 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. $\mathcal{H}_{7}$-tetrads of lines in $\operatorname*{PG}(7,2)$
  3. $\mathcal{G}(\mathcal{L}_{4})$-invariant polynomials
  4. The eight distinguished spreads $\{\mathcal{L}_{85}^{ijk}\}_{i,j,k\in\{1,2\}}$
  5. The normal subgroup $\mathcal{G}_{81}$ of $\mathcal{G}(\mathcal{L}_{4})$
  6. A $\operatorname*{GF}(3)$ view of $\mathcal{G}_{81}$
  7. The 27-set 'denizens' of $\omega_{4}$
  8. Non-Segre triplets in $\omega_{4}$
  9. Intersection properties
  10. Introduction
  11. Sections of a Segre variety $\mathcal{S}\subset\omega_{4}$
  12. Hamming distances and troikas
  13. Future research

Key Result

Theorem 1

The quadric $\mathcal{Q}$ is a hyperbolic quadric $\mathcal{H}_{7};$ moreover $\mathcal{H}_{7}=\omega_{2}\cup\omega_{4}.$

Theorems & Definitions (41)

  • Theorem 1
  • proof
  • Corollary 2
  • Remark 3
  • Theorem 4
  • proof
  • Remark 5
  • Remark 6
  • Lemma 7
  • proof
  • ...and 31 more