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
Definition 2.2 (SU3). Let SU3 be the group of collineations of
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 ∩
where π is the hyperplane X3 = 0
and let GO be the stabiliser of O in PGU4(q). Let b be a Baer
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
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
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
Let ω be an element of
satisfying N(ω) = −1, and let U0 and
0 0 −ω
0 0 1
:= (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
g gω 1
−f −fω b−aω
and we obtain
(k−1−k−bωq)ωq k+bωq k−1gω
g gω 1
where k ∈ GF(q), N(g) =
− 1 and T(gω) = 0 (in order for this
matrix to be unitary).
The determinant of A is
+ b(N(ω) +
+ ω) = 1
and therefore, the stabiliser of [U0, 0] in GO is contained in SU3.