Symplectic polarities of buildings of type E-6
A symplectic polarity of a building Delta of type E (6) is a polarity whose fixed point structure is a building of type F (4) containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality theta of Delta never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then theta is a symplectic duality. Secondly, we show that, if a duality theta never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E (6) of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E (6) for which the Phan geometry is empty.
domestic automorphisms spherical buildings dualities of buildings of type e-6