Abstract
We describe how to obtain the projective coordinates of the points of the Ree–Tits generalized octagon over a perfect field K in the 25-dimensional embedding of [K. Coolsaet, Algebraic structure of the perfect Ree-Tits generalized octagons. Innov. Incidence Geom. 1 (2005), 67–131. MR2213954 (2007a:51004) Zbl 05031723] from their Van Maldeghem coordinates (as defined in [H. Van Maldeghem, Generalized polygons. Birkhäuser 1998. MR1725957 (2000k:51004) Zbl 0914.51005, Section 3.6]). As an important bonus the explicit formulas which result from this may now serve as the definition of a 25-dimensional embedding also in the non-perfect case. We also give explicit formulas for the operations Ψ i and Φ j of the octagonal octanary ring associated with the octagon and its dual. (The latter are new.)
As an application we derive the dimensions of subspaces of this embedding generated by various special subsets of points of the octagon: the sets of points at a fixed distance from a given point or a given line, the Suzuki suboctagons, the traces and the hypertraces. In some of the cases these dimensions turn out to depend on whether K is the field of 2 elements, or not.
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