Geometries, the principle of duality, and algebraic groups

by Michael Carr and Skip Garibaldi

Expositiones Mathematicae 24 (2006) 195-234

Abstract: Jacques Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits's geometry and works through several examples. We also give a criterion for recognizing the automorphism of the geometry induced by an automorphism of the group. The E₆ geometry is studied in depth.

This paper replaces an old, unpublished note titled Twisted flag varieties of exceptional algebraic groups.

Warning: In the published version, there is a typo in Figure 1 on page 209. The weight at the bottom of the rightmost column should be (-1, -1, 1, 0, 0, 0) and not (-1, -1, 1, 1, 0, 0, 0). (Hopefully even the most casual reader would notice that there is a typo, since each vector should have only 6 entries, and the table should be symmetric with respect to the map -phi.)

Also, the authors are reversed. We should be listed in alphabetical order, with Carr first.

Version of 30 September 2005. This is essentially the published version.
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For an alternative approach to describing the projective homogeneous varieties---hence also the geometries---associated with groups of type A and D, see Pat Morandi's unpublished note Algebraic groups, Grassmannians, and flag varieties.