On Embeddings of the Flag Geometries of Projective Planesin Finite Projective Spaces

  • Authors:

  • Joseph A. Thas;Hendrik Van Maldeghem

  • Affiliations:

  • Department of Pure Mathematics and Computer Algebra, University Ghent, Galglaan 2, B – 9000 Gent;Department of Pure Mathematics and Computer Algebra, University Ghent, Galglaan 2, B – 9000 Gent

  • Venue:
  • Designs, Codes and Cryptography - Special issue on designs and codes—a memorial tribute to Ed Assmus
  • Year:
  • 1999

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Abstract

The flaggeometry \Gamma=(\cP,\cL,\op) of a finite projectiveplane \Pi of order s is the generalizedhexagon of order (s,1) obtained from \Piby putting \cP equal to the set of all flags of\Pi, by putting \cL equal to the setof all points and lines of \Pi and where \opis the natural incidence relation (inverse containment), i.e.,\Gamma is the dual of the double of \Piin the sense of Van Maldeghem Mal:98. Then we say that \Gammais fully and weakly embedded in the finite projective space \PG(d,q)if \Gamma is a subgeometry of the natural point-linegeometry associated with \PG(d,q), if s=q,if the set of points of \Gamma generates \PG(d,q),and if the set of points of \Gamma not oppositeany given point of \Gamma does not generate \PG(d,q).Preparing the classification of all such embeddings, we constructin this paper the classical examples, prove some generalitiesand show that the dimension d of the projectivespace belongs to \{6,7,8\}.