J. Korean Math. Soc. 2023; 60(4): 907-929
Online first article June 12, 2023 Printed July 1, 2023
https://doi.org/10.4134/JKMS.j220528
Copyright © The Korean Mathematical Society.
Sebastian Petit, Hendrik Van Maldeghem
Krijgslaan 281--S25; Krijgslaan 281--S25
We consider thick generalized hexagons fully embedded in metasymplectic spaces, and we show that such an embedding either happens in a point residue (giving rise to a full embedding inside a dual polar space of rank 3), or happens inside a symplecton (giving rise to a full embedding in a polar space of rank 3), or is isometric (that is, point pairs of the hexagon have the same mutual position whether viewed in the hexagon or in the metasymplectic space--these mutual positions are \emph{equality, collinearity, being special, opposition}). In the isometric case, we show that the hexagon is always a Moufang hexagon, its little projective group is induced by the collineation group of the metasymplectic space, and the metasymplectic space itself admits central collineations (hence, in symbols, it is of type $\mathsf{F_{4,1}}$). We allow non-thick metasymplectic spaces without non-thick lines and obtain a full classification of the isometric embeddings in this case.
Keywords: Moufang hexagon, full embedding, metasymplectic spaces, equator geometry
MSC numbers: Primary 51E24, 51E12
Supported by: The second author is partially supported by the Fund for Scientific Research Flanders through the grant G023121N.
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