Geometries, the principle of duality, and algebraic groups

Abstract

Jacques Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits's geometry and works through several examples. We also give a criterion for recognizing the automorphism of the geometry induced by an automorphism of the group. The E6 geometry is studied in depth.