Algebraic structure of the perfect Ree-Tits generalized octagons

Abstract

We provide an algebraic description of the perfect Ree-Tits generalized octagons, i.e., an explicit embedding of octagons of this type in a 25-dimensional projective space. The construction is derived from the interplay between the 52-dimensional module of the Chevalley algebra of type ${\mathsf{F}}_4$ over a field of even characteristic and its 26-dimensional submodule. We define a quadratic duality operator that interchanges special sets of (totally) isotropic elements in those modules and establish the points of the octagon as absolute points of this duality. We introduce many algebraic operations that can be used in the study of the generalized octagon. We also prove that the Ree group acts as expected on points and pairs of points.