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 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 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 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.
Previous Page Next Page