LOW DIMENSIONAL MODELS OF THE FINITE SPLIT CAYLEY HEXAGON 7

The key to this construction is the action of a particular subgroup of

GO. We will see later that this group naturally corresponds to the stabiliser

in G2(q) of a non-degenerate hyperplane

Q−(5,q)

of Q(6,q).

Definition 2.2 (SU3). Let SU3 be the group of collineations of

H(3,q2)

obtained from the matrices MA where A ∈ SU3(q).

In short, the orbits of SU3 on Baer subgenerators with a point in O,

each form a suitable candidate for Ω, as we will see.

Lemma 2.3. Let O = π ∩

H(3,q2),

where π is the hyperplane X3 = 0

of

PG(3,q2),

and let GO be the stabiliser of O in PGU4(q). Let b be a Baer

subgenerator of

H(3,q2)

with a point in O. Then the stabiliser of b in GO is

contained in SU3.

Proof. Recall from the beginning of Section 2 that given a Baer sub-

generator b of

H(3,q2)

with a point B in O, there is a dual Baer subline of

π with vertex B. So there is a set of 3 × 3 Hermitian matrices U of rank

2, which are equivalent up to scalar multiplication in

GF(q2)∗.

Now GO

induces an action on the pairs [U, ], where U is a Hermitian matrix of rank

2 and is a generator containing the nullspace of U, which we can write out

explicitly by

[U,

]MA

=

[A−1UA, MA

].

Let ω be an element of

GF(q2)

satisfying N(ω) = −1, and let U0 and

0

be

U0 :=

0 0 −ω

0 0 1

−ωq

1 0

,

0

:= (1,ω, 0, 0), (0, 0, 1,ω) .

Since GO acts transitively on Baer subgenerators with a point in O (Lemma

2.1), we need only calculate the stabiliser of [U0, 0]. Now let MA be an ele-

ment of GO fixing [U0, 0]. Since MA fixes 0, we can see by direct calculation

that A is of the form

a b −fω

d e f

g gω 1

,

with (a + dω)ω = b + eω.

Now we see what it means for A to centralise U0 up to a scalar k, that is,

U0A = kAU0. Hence

−gω

−gω2

−ω

g gω 1

d−aωq e−bωq

0

= k

−f −fω b−aω

−fωq

f e−dω

−ωq

1 0

and we obtain

A =

k−1−bωq

b

−k−1gω2

(k−1−k−bωq)ωq k+bωq k−1gω

g gω 1

where k ∈ GF(q), N(g) =

k2

+

T(bqω)

− 1 and T(gω) = 0 (in order for this

matrix to be unitary).

The determinant of A is

1 −

g2ω(N(ω)

+

1)(ωk−2

+ b(N(ω) +

1)k−1

+ ω) = 1

and therefore, the stabiliser of [U0, 0] in GO is contained in SU3.