Innovations in Incidence Geometry

Algebraic structure of the perfect Ree-Tits generalized octagons

Kris Coolsaet

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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.

Article information

Innov. Incidence Geom., Volume 1, Number 1 (2005), 67-131.

Received: 13 January 2005
Accepted: 13 March 2005
First available in Project Euclid: 26 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B25: Exceptional (super)algebras 51E12: Generalized quadrangles, generalized polygons

Ree-Tits generalized octagon Chevalley algebra of type F4


Coolsaet, Kris. Algebraic structure of the perfect Ree-Tits generalized octagons. Innov. Incidence Geom. 1 (2005), no. 1, 67--131. doi:10.2140/iig.2005.1.67.

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