Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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GENERALIZED HEXAGONS EMBEDDED IN METASYMPLECTIC SPACES

  • Received : 2022.10.27
  • Accepted : 2023.05.17
  • Published : 2023.07.01
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Abstract

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 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 F4,1). We allow non-thick metasymplectic spaces without non-thick lines and obtain a full classification of the isometric embeddings in this case.

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Acknowledgement

The authors warmly thank Anneleen De Schepper and Linde Lambrecht for some fruitful discussions on the topic of the paper. They would also like to thank the anonymous referee for a careful reading and inquiring some more explanation at various points that made the paper more accessible.

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