A 51-dimensional embedding of the Ree-Tits generalized octagon
In his PhD thesis, Kris Coolsaet has developed a marvelous theory around an alternative algebraic construction of the Ree-Tits generalized octagons in the perfect case. In this theory, a 25-dimensional projective embedding of these Moufang octagons arises. The existence of this embedding, however, is not a surpise, as the author points out himself. Indeed, such an embedding was previously known to exist since the octagons are embedded in a metasymplectic space, and the metasymplectic space is embedded in a 25-dimensional projective space. The surprising fact is that Coolsaet can easily give a complete, direct and explicit description of this embedding in terms of the very concrete ``Van Maldeghem coordinates'' (with the author's terminology, but from now on simply called ``polygonal coordinates''). These coordinates, in turn, are derived directly from the commutation relations of the root groups. Hence, one could say that the 25-dimensonal embedding has a direct description using the Ree groups. The same now holds for a 51-dimensional embedding. Indeed, it is known that the dual metasymplectic spaces are embeddable in 51-dimensional projective spaces, and since the octagons are also embedded in the dual metasymplectic spaces (since they arise from a polarity), they also embed in 51-dimensional projective spaces. Now, using his theory, Coolsaet ingeniously derives a simple algebraic construction of this embedding, and then translates everything in polygonal coordinates. The advantage is that we can now forget about the algebra, keep the coordinates and drop the assumption on the perfectness of the underlying field. Consequently the author obtains a similar embedding for all Moufang octagons (perfect and non-perfect Ree-Tits octagons). Finally, the author provides some information on the dimension of subspaces generated by special point sets of the Ree-Tits octagons.
scientific article; zbMATH DE number 5536720
scientific article