LOW DIMENSIONAL MODELS OF THE FINITE SPLIT CAYLEY HEXAGON 3
Moreover, Γ is isomorphic to the split Cayley hexagon H(q).
The proof that Γ is a generalised hexagon is presented in Section 2.1.
Note that the lines of type (i) form a spread of H(q). There exists a natural
candidate for Ω which we explain in detail in Section 2.2, and it is essentially
the only one (Theorem 2.6), and this implies the ultimate result that Γ is
isomorphic to H(q).
By the deep results of Thas and Van Maldeghem [18, 19] and Cameron
and Kantor [6], if a set of points P and lines L of PG(6,q) form a generalised
hexagon, then it is isomorphic to the split Cayley hexagon H(q) if P spans
PG(6,q) and for any point x P, the points collinear to x span a plane.
A similar result was proved recently by Thas and Van Maldeghem [20], by
foregoing the assumption that P and L form a generalised hexagon, and
instead instituting the following five axioms: (i) the size of L is
(q6
−1)/(q
1), (ii) every point of PG(6,q) is incident with either 0 or q + 1 elements of
L, (iii) every plane of PG(6,q) is incident with 0, 1 or q + 1 elements of L,
(iv) every solid of PG(6,q) contains 0, 1, q + 1 or 2q + 1 elements of L, and
(v) every hyperplane of PG(6,q) contains at most
q3
+
3q2
+ 3q elements of
L.
One could instead characterise the split Cayley hexagon viewed as points
and lines of the parabolic quadric Q(6,q), and the best result we have to
date follows from a result of Cuypers and Steinbach [7, Theorem 1.1]:
Theorem 1.3 (Cuypers and Steinbach [7] (paraphrased)). Let L be a
set of lines of Q(6,q) such that every point of Q(6,q) is incident with q + 1
lines of L spanning a plane, and such that the concurrency graph of L is
connected. Then the points of Q(6,q) together with L define a generalised
hexagon isomorphic to the split Cayley hexagon H(q).
In Section 4 we will give an elementary proof of Theorem 1.3 by using
Theorem 2.6.
Some remarks on notation: In this paper, the relative norm and
relative trace maps will be defined for the quadratic extension
GF(q2)
over
GF(q). The relative norm N is the multiplicative function which maps an
element x
GF(q2)
to the product of its conjugates of
GF(q2)
over GF(q).
That is, N(x) =
xq+1.
The relative trace is instead the sum of the conjugates,
T(x) := x +
xq.
2. The 3-dimensional Hermitian surface and its Baer
substructures
The two (classical) generalised quadrangles of particular importance in
this note are
H(3,q2)
and
Q−(5,q).
First there is the incidence structure of
all points and lines of a non-singular Hermitian variety in
PG(3,q2),
which
forms the generalised quadrangle
H(3,q2)
of order
(q2,q).
Its point-line dual
is isomorphic to the geometry of points and lines of the elliptic quadric
Q−(5,q)
in PG(5,q), which yields a generalised quadrangle of order (q,
q2)
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