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

Sebastian Petit, Geertrui Van de Voorde

TL;DR

This work characterises the natural embedding of the twisted triality hexagon $\mathsf{T}(q^3,q)$ into $\mathsf{PG}(7,q^3)$ by analyzing how subspaces meet the embedded hexagon and by formulating a line-set framework with properties (Pt),(Pl),(Sd),(4d),(4d'),(5d),(6d) and (To). It introduces distance-3 traces and regulus structures to control intersection patterns and proves that a line set satisfying these criteria corresponds to a regularly embedded $\mathsf{T}(q^3,q)$, mirroring the approach used for the split Cayley hexagon. The results provide a rigorous characterization of the natural embedding via line sets and subspace interactions, building a bridge between incidence geometry of generalized hexagons and projective embeddings. The findings extend prior work on split Cayley hexagons to twisted triality hexagons and establish a robust framework for recognizing and constructing such embeddings in $\mathsf{PG}(7,q^3)$ with potential implications for finite geometry and related combinatorial structures.

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).

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

TL;DR

This work characterises the natural embedding of the twisted triality hexagon into by analyzing how subspaces meet the embedded hexagon and by formulating a line-set framework with properties (Pt),(Pl),(Sd),(4d),(4d'),(5d),(6d) and (To). It introduces distance-3 traces and regulus structures to control intersection patterns and proves that a line set satisfying these criteria corresponds to a regularly embedded , mirroring the approach used for the split Cayley hexagon. The results provide a rigorous characterization of the natural embedding via line sets and subspace interactions, building a bridge between incidence geometry of generalized hexagons and projective embeddings. The findings extend prior work on split Cayley hexagons to twisted triality hexagons and establish a robust framework for recognizing and constructing such embeddings in with potential implications for finite geometry and related combinatorial structures.

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).

Paper Structure

This paper contains 21 sections, 61 theorems, 28 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Generalised hexagons
  4. The standard embedding of $\mathsf{T}(q^3,q)$ in $\mathsf{PG}(7,q^3)$
  5. The standard embedding of $\mathsf{T}(q^3,q)$ in $\mathsf{PG}(7,q^3)$ satisfies the properties
  6. Case 1.
  7. Case 2.
  8. Case 1: $L$ does not intersect $\pi$.
  9. Case 2: $L$ intersects $\pi$.
  10. Case 1.a: $|\mathcal{L}^{\mathsf{T}}(U)| = q^2+q+1$ and $L$ intersects $M$.
  11. Case 1.b: $|\mathcal{L}^{\mathsf{T}}(U)| = q^2+q+1$ and $L$ does not intersect $M$.
  12. Case 2.a: $|\mathcal{L}^{\mathsf{T}}(U)| = q^2+2q+1$ and $L$ intersects $\pi$.
  13. Case 2.b: $|\mathcal{L}^{\mathsf{T}}(U)| = q^2+2q+1$ and $L$ does not intersect $\pi$.
  14. Case 1.a: $x \not\in L_2$, $c\not \in M$.
  15. Case 1.b: $x \not\in L_2$, $c \in M$.
  16. ...and 6 more sections

Key Result

Theorem 1

Figures (10)

  • Figure 1: Coordinates
  • Figure 2: Case 1
  • Figure 3: Case 2
  • Figure 4: Case 1.a
  • Figure 5: Case 1.b
  • ...and 5 more figures

Theorems & Definitions (127)

  • Theorem 1: Thas & Van Maldeghem ThasVanMaldeghem1998
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Definition 2
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Definition 3
  • ...and 117 more