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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