Symplectic polarities of buildings of type E 6

A symplectic polarity of a building Δ 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 θ of Δ never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then θ is a symplectic duality. Secondly, we show that, if a duality θ 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.