Vol. 1, No. 1, 2005

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Algebraic structure of the perfect Ree-Tits generalized octagons

Kris Coolsaet

Vol. 1 (2005), No. 1, 67–131

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 F4 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.

Ree-Tits generalized octagon, Chevalley algebra of type F4
Mathematical Subject Classification 2000
Primary: 17B25, 51E12
Received: 13 January 2005
Accepted: 13 March 2005
Kris Coolsaet
Vakgroep Toegepaste Wiskunde en Informatica
Universiteit Gent
Krijgslaan 281-S9
9000 Gent