Abstract

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 $F_4$ 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 $F_4$ 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.

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