Symplectic polarities of buildings of type E 6 |
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Authors: | Hendrik Van Maldeghem |
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Affiliation: | 1. Department of Mathematics, Ghent University, Krijgslaan 281-S22, 9000, Ghent, Belgium
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Abstract: | 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. |
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