Let for some field and be the subgroup of upper triangular matrices in . As usual, let denote the projective plane over .

As -varieties, is isomorphic to because is the stabilizer of the usual (Mobius) action of on . The resulting action of on is 1-transitive and fixed-point free. Indeed, counting points when is a finite field with elements agrees with the facts that

and

.

This generalizes in that parabolic subgroups (closed subgroup containing a Borel subgroup, i.e., containing the upper triangular matrices upto conjugation, for the general linear group) of an algebraic group give rise to projective varieties. They are precisely the stabilizers of a flag and conversely, the set of flags can be given the structure of an -space and in particular a topology wherein they are compact spaces (being projective spaces). They are called flag varieties.

## Leave a comment

Comments feed for this article