Buildings of exceptional type in buildings of type $\mathsf{E_7}$
Volume 573 / 2022
Abstract
We investigate the possible ways in which a thick metasymplectic space $\Gamma$, that is, a Lie incidence geometry of type $\mathsf{F_{4,1}}$ (or $\mathsf{F_{4,4}}$), is embedded into the long root geometry $\Delta$ related to any building of type $\mathsf{E_7}$. We provide a complete classification (if $\Gamma$ is not embedded in a singular subspace). As an application we prove the uniqueness of the inclusion of the long root geometry of type $\mathsf{E_6}$ in the one of type $\mathsf{E_7}$; it always arises as an equator geometry. We also use the latter concept to geometrically construct one of the embeddings turning up in our classification. As a side result we find that all triples of pairwise opposite elements of type $7$ in a building of type $\mathsf{E_7}$ are projectively equivalent.
