A distance-2-spread of the generalized hexagon H(3)
A distance-2-spread of a generalized hexagon \(\Gamma\) is a set \(\mathcal S\) of non-concurrent lines of \(\Gamma\) such that every point of \(\Gamma\) is incident with exactly one line of \(\mathcal S\); note that the concepts ``distance-2-spread of \(\Gamma\)'' and ``spread of \(\Gamma\)'' differ; cf. chapters 7.2.1 and 7.3.9 of \textit{H. Van Maldeghem} [Generalized polygons (Monographs in Mathematics. 93. Basel: Birkhäuser) (1998; Zbl 0914.51005)]. A distance-2-ovoid of \(\Gamma\) is the dual of a distance-2-spread of \(\Gamma\). The point set of the split Cayley hexagon \({\mathbb H}(q)\) coincides with the point set of the quadric \({\mathbb Q}(6,q)\) of \({\mathbb PG}(6,q)\) having the equation \(X_{0}X_4+X_{1}X_5+X_{2}X_6=X_{3}^2\) and the line set of \({\mathbb H}(q)\) is a certain proper subset of the lines on \({\mathbb Q}(6,q)\); cf. p.110 of op. cit. A set of lines which partitions \({\mathbb Q}(6,q)\) is called a line spread of \({\mathbb Q}(6,q)\). Each distance-2-spread of \({\mathbb H}(q)\) is a line spread of \({\mathbb Q}(6,q)\). The authors construct a line spread of \({\mathbb Q}(6,2)\) and show that \({\mathbb H}(2)\) admits no distance-2-spread. The authors prove that \({\mathbb H}(3)\) contains, up to isomorphisms of \({\mathbb H}(3)\) a unique distance-2-spread (and by self-duality of \({\mathbb H}(3)\) a unique distance-2-ovoid). Its automorphism group \(G\) is isomorphic to the projective special linear group \textbf{PSL}\(_2(13)\) which is a maximal subgroup of the automorphism group \({\mathbb G}_2(3)\) of \({\mathbb H}(3)\). The uniqueness proof uses the classification of flag matchings of \textbf{PG}\((2,3)\); every set of \(1+n+n^2\) flags of a projective plane \(\Pi\) of order \(n\) which cover all points and all lines of \(\Pi\) is called a flag matching of \(\Pi\). Up to isomorphisms there exist exactly five flag matchings of \textbf{PG}\((2,3)\). By application of the above results the authors prove also: 1. The chromatic number of the point graph of \({\mathbb H}(3)\) is larger than \(4\). 2. The twisted triality hexagons \({\mathbb T}(8,2)\) and \({\mathbb T}(27,3)\) do not contain any distance-2-spread.
scientific article; zbMATH DE number 2103719