Daan Rijpert1, Hendrik Van Maldeghem1
1Department of Mathematics, Computer Science and Statistics, Ghent University, Belgium
Abstract:

Recently, the notion of a Weyl substructure in a spherical building was introduced type by type. In this paper we provide a uniform (axiomatic) definition across all types. In particular, this provides a new characterisation of the Ree-Tits octagons. We then show that uniclass automorphisms of spherical buildings are uniformly characterised by their fix structure. For type preserving automorphisms, this follows from earlier work, and so the focus here is on dualities. In particular, it follows that a duality pointwise fixing a Weyl substructure is automatically a polarity. This characterises all polarities in self-dual spherical buildings where opposition acts trivially on the types.