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Mixed relations for buildings of type $\mathsf{F_4}$

Johannes Roth and Hendrik Van Maldeghem

Vol. 20 (2023), No. 2-3, 543–578

The main result of this paper is an explicit description of the representation of the metasymplectic space related to an arbitrary building of mixed type F4 in 25-dimensional projective space. As an application, we study collineations of such spaces the fixed point structure of which is a Moufang quadrangle. We show that the exceptional Moufang quadrangles of type F4 can be obtained as the intersection of the mixed metasymplectic space with a Baer subspace of the ambient projective space. We also determine the group of collineations fixing a mixed quadrangle and, more surprisingly, observe that it has infinite order, whereas it was generally believed to have just order 2. Finally, we classify collineations of the mixed metasymplectic space fixing mixed Moufang quadrangles arising from subspaces.

Tits-buildings, metasymplectic spaces, Moufang quadrangles, mixed groups
Mathematical Subject Classification
Primary: 51E24
Received: 3 May 2022
Revised: 16 March 2023
Accepted: 16 May 2023
Published: 13 September 2023
Johannes Roth
Department of Mathematics: Algebra and Geometry
Ghent University
Hendrik Van Maldeghem
Department of Mathematics: Algebra and Geometry
University of Ghent