19 - Generalized hexagons as geometric hyperplanes of near hexagons

Summary

Introduction.

The first theorem given here asserts that a geometric hyperplane H of a near hexagon, which intersects each quad at a star must be a generalized hexagon. The second theorem tells us that if a finite near hexagon with parameters possesses such a geometric hyperplane, then that near hexagon Γ must be the dual of a rank 3 polar space Δ. Moreover, there is a bijection H ↔ quads of Γ, which induces an embedding of the hexagon H into the polar space A which is an epimorphism on points. Conceivably, there is a possibility that generalized hexagons might be represented as geometric hyperplanes of some of the “other” dual polar spaces, such as Ω(n, ℝ) (with signature (n – 3,3)), Sp(6, κ), Ω⁻(8, κ), U(6, κ) or U(7, κ). But the final theorem shows that if Γ is finite, such possibilities cannot happen; that in fact Γ (and Δ) are type Ω(7, q) (or Sp(6, q) if q is even) and H is the hexagon of type G₂(q) associated with the standard embedding of G₂(q) (either as the stabilizer of an appropriate hyperplane in the 8-dimensional spin module for Ω(7, q) or as the stabilizer of a trilinear form in its natural 7-dimensional module – or the factor of this 7-space module by a 1-dimensional radical when q is even).

The author thanks Professor J. Tan for a valuable discussion, Queen Mary College, U. of London, and the Mathematisches Institute, Albert-Ludwigs Universität Freiburg for their kind hospitality during the writing of this work, and the Alexander von Humboldt Stiftung whose support made the research possible.