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Automorphisms and opposition in spherical buildings of exceptional type, III : metasymplectic spaces

Linde Lambrechts and Hendrik Van Maldeghem (UGent)
(2025) SELECTA MATHEMATICA-NEW SERIES. 32(1).
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Abstract
We classify all domestic collineations, that is, collineations mapping no chamber to an opposite one, of all thick spherical buildings of type F-4. Besides the previously known cases like central elations and products of two perpendicular such elations, we find collineations that pointwise fix certain subspaces, also of type F-4, but over a smaller algebra, or even non-thick as a building. We also find examples that pointwise fix Moufang quadrangles, and these inclusions are new: Moufang quadrangles of absolute type D-5 are contained in buildings of type F-4 of absolute type E-6, and exceptional Moufang quadrangles of type E-6 are found inside buildings of relative type F-4 and absolute type E-7 (the so-called quaternion metasymplectic spaces). Together with the already established Moufang quadrangles of mixed type inside mixed buildings of type F-4, our results imply that domestic collineations give rise to inclusions of the three different types of Moufang quadrangles inside metasymplectic spaces: Moufang quadrangles of classical, exceptional and mixed type.
Keywords
Exceptional spherical buildings of type F-4, Opposition diagram, Domestic automorphism, Moufang quadrangles, POLAR, QUADRANGLES

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MLA
Lambrechts, Linde, and Hendrik Van Maldeghem. “Automorphisms and Opposition in Spherical Buildings of Exceptional Type, III : Metasymplectic Spaces.” SELECTA MATHEMATICA-NEW SERIES, vol. 32, no. 1, 2025, doi:10.1007/s00029-025-01111-z.
APA
Lambrechts, L., & Van Maldeghem, H. (2025). Automorphisms and opposition in spherical buildings of exceptional type, III : metasymplectic spaces. SELECTA MATHEMATICA-NEW SERIES, 32(1). https://doi.org/10.1007/s00029-025-01111-z
Chicago author-date
Lambrechts, Linde, and Hendrik Van Maldeghem. 2025. “Automorphisms and Opposition in Spherical Buildings of Exceptional Type, III : Metasymplectic Spaces.” SELECTA MATHEMATICA-NEW SERIES 32 (1). https://doi.org/10.1007/s00029-025-01111-z.
Chicago author-date (all authors)
Lambrechts, Linde, and Hendrik Van Maldeghem. 2025. “Automorphisms and Opposition in Spherical Buildings of Exceptional Type, III : Metasymplectic Spaces.” SELECTA MATHEMATICA-NEW SERIES 32 (1). doi:10.1007/s00029-025-01111-z.
Vancouver
1.
Lambrechts L, Van Maldeghem H. Automorphisms and opposition in spherical buildings of exceptional type, III : metasymplectic spaces. SELECTA MATHEMATICA-NEW SERIES. 2025;32(1).
IEEE
[1]
L. Lambrechts and H. Van Maldeghem, “Automorphisms and opposition in spherical buildings of exceptional type, III : metasymplectic spaces,” SELECTA MATHEMATICA-NEW SERIES, vol. 32, no. 1, 2025.
@article{01KRX3TNAQAK7M3GCZM7KD4PWK,
  abstract     = {{We classify all domestic collineations, that is, collineations mapping no chamber to an opposite one, of all thick spherical buildings of type F-4. Besides the previously known cases like central elations and products of two perpendicular such elations, we find collineations that pointwise fix certain subspaces, also of type F-4, but over a smaller algebra, or even non-thick as a building. We also find examples that pointwise fix Moufang quadrangles, and these inclusions are new: Moufang quadrangles of absolute type D-5 are contained in buildings of type F-4 of absolute type E-6, and exceptional Moufang quadrangles of type E-6 are found inside buildings of relative type F-4 and absolute type E-7 (the so-called quaternion metasymplectic spaces). Together with the already established Moufang quadrangles of mixed type inside mixed buildings of type F-4, our results imply that domestic collineations give rise to inclusions of the three different types of Moufang quadrangles inside metasymplectic spaces: Moufang quadrangles of classical, exceptional and mixed type.}},
  articleno    = {{8}},
  author       = {{Lambrechts, Linde and Van Maldeghem, Hendrik}},
  issn         = {{1022-1824}},
  journal      = {{SELECTA MATHEMATICA-NEW SERIES}},
  keywords     = {{Exceptional spherical buildings of type F-4,Opposition diagram,Domestic automorphism,Moufang quadrangles,POLAR,QUADRANGLES}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{102}},
  title        = {{Automorphisms and opposition in spherical buildings of exceptional type, III : metasymplectic spaces}},
  url          = {{http://doi.org/10.1007/s00029-025-01111-z}},
  volume       = {{32}},
  year         = {{2025}},
}
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