Intersection numbers of the natural embedding of the twisted triality hexagon T(q^3,q) in PG(7,q^3)

TL;DR

This paper characterizes the intersection properties of the natural embedding of the twisted triality hexagon T(q^3,q) in PG(7,q^3), providing conditions for line sets to form such a hexagon, extending previous work on related structures.

Contribution

It offers a detailed description of intersections and conditions for embedding twisted triality hexagons in projective space, advancing understanding of their geometric properties.

Findings

Characterization of subspace intersections with T(q^3,q)

Conditions for line sets to form embedded hexagons

Extension of previous results on split Cayley hexagon

Abstract

In this paper, we study and characterise the natural embedding of the twisted triality hexagon T(q^3,q) in PG(7,q^3). We begin by describing the possible intersections of subspaces of PG(7,q^3) with T(q^3,q). Then, we provide conditions on a set of lines L which ensures that L forms the line set of a naturally embedded twisted triality hexagon. This work follows up on similar results for the split Cayley hexagon by J. A. Thas and H. Van Maldeghem (2008) and F. Ihringer (2014).