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30 - Embedding the root group geometry of 2F4(q)

Published online by Cambridge University Press:  05 March 2012

J. Sarli
Affiliation:
California State University, U.S.A.
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Summary

In the standard interpretation of the 2-local geometry of a group of Lie type defined over a field of even characteristic it is well-known that the geometry of the finite Ree groups of type F4, is that of a generalized octagon with parameters (q,q2), q the order of the field. Outside of the theory of buildings itself this is perhaps most easily seen within an irreducible Fq-module of minimal dimension where each maximal 2-local P is identified with the subspace centralized by 02(P). It is also well-known that this geometry can be realized as a point-line incidence structure inside the group by taking the centers of the non-abelian root groups as the points and the groups generated by the pairs of such centers in a distinr guished orbital as the lines. (This latter approach, of Cooperstein, was the subject of a considerable body of work on the long root groups of exceptional Lie type, [4].) An internal refinement (without reference to a representation space) has been given in [8], explaining the roles of the various root groups (there are three root lengths to consider in a non reduced system; cf. Tits [11]) and resulting in a configuration built on the octagon which resembles a metasymplectic space. The purpose here is to give a direct description of the fundamental elements of this geometry inside a 26-dimensional Fq-space for these groups. Section 1 gives some terse background and suggests a basis for the module which minimizes calculations.

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Publisher: Cambridge University Press
Print publication year: 1987

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