Innovations in Incidence Geometry

Algebraic structure of the perfect Ree-Tits generalized octagons

Kris Coolsaet

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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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551206816

Digital Object Identifier
doi:10.2140/iig.2005.1.67

Mathematical Reviews number (MathSciNet)
MR2213954

Zentralblatt MATH identifier
1102.51002

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

Keywords
Ree-Tits generalized octagon Chevalley algebra of type F4

Citation

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. https://projecteuclid.org/euclid.iig/1551206816


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References

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