Contemporary Mathematics
Volume 579, 2012
http://dx.doi.org/10.1090/conm/579/11514
Low dimensional models of the finite split Cayley
hexagon
John Bamberg and Nicola Durante
Abstract. We provide a model of the split Cayley hexagon arising from
the Hermitian surface H(3,q2), thereby yielding a geometric construction
of the Dickson group G2(q) starting with the unitary group SU3(q).
1. Introduction
A generalised polygon Γ is a point-line incidence structure such that the
incidence graph is connected and bipartite with girth twice that of its diam-
eter. If the valency of every vertex is at least 3, then we say that Γ is thick,
and it turns out that the incidence graph is then
biregular1.
By a famous
result of Feit and Higman [8], a finite thick generalised polygon is a com-
plete bipartite graph, projective plane, generalised quadrangle, generalised
hexagon or generalised octagon. There are many known classes of finite
projective planes and finite generalised quadrangles but presently there are
only two known families, up to isomorphism and duality, of finite generalised
hexagons; the split Cayley hexagons and the twisted triality hexagons.
The split Cayley hexagons H(q) are the natural geometries for Dickson’s
group G2(q), and they were introduced by Tits [21] as the set of points of the
parabolic quadric Q(6,q) and an orbit of lines of Q(6,q) under G2(q). If q is
even, then the polar spaces W(5,q) and Q(6,q) are isomorphic geometries,
and hence H(q) can be embedded into a five-dimensional projective space.
Thas and Van Maldeghem [19] proved that if H is a finite thick generalised
hexagon
embedded2
into the projective space PG(d, q), then d 7 and
this embedding is equivalent to one of the standard models of the known
generalised hexagons. So in particular, it is impossible to embed the split
Cayley hexagon H(q) into a three-dimensional projective space. However,
2010 Mathematics Subject Classification. Primary 05B25, 51E12, 51E20.
Key words and phrases. Generalised hexagon, Hermitian surface.
1That is, there are two constants k1 and k2 such that the valency of each vertex in
one bipartition is k1, and the valency of each vertex in the other bipartition is k2.
2We
will not discuss the various meanings of “embedding” here, but instead refer the
interested reader to [18, 19].
c 2012 American Mathematical Society
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