Low dimensional models of the finite split Cayley hexagon

TL;DR

This paper constructs a geometric model of the split Cayley hexagon using Hermitian surfaces, linking the Dickson group G_2(q) with the unitary group SU_3(q) through a novel geometric approach.

Contribution

It introduces a new geometric construction of the split Cayley hexagon from Hermitian surfaces, connecting algebraic groups with geometric models.

Findings

Established a geometric model of the split Cayley hexagon from Hermitian surfaces.

Derived a construction of the Dickson group G_2(q) from the unitary group SU_3(q).

Provided insights into the relationship between algebraic groups and finite geometries.

Abstract

We provide a model of the split Cayley hexagon arising from the Hermitian surface $\mathsf{H}(3,q^2)$, thereby yielding a geometric construction of the Dickson group $G_2(q)$ starting with the unitary group $\mathsf{SU}_3(q)$.