Generalized Flag Variety - Prototype: The Complete Flag Variety

Prototype: The Complete Flag Variety

According to basic results of linear algebra, any two complete flags in an n-dimensional vector space V over a field F are no different from each other from a geometric point of view. That is to say, the general linear group acts transitively on the set of all complete flags.

Fix an ordered basis for V, identifying it with Fn, whose general linear group is the group GL(n,F) of n × n matrices. The standard flag associated with this basis is the one where the i th subspace is spanned by the first i vectors of the basis. Relative to this basis, the stabilizer of the standard flag is the group of nonsingular upper triangular matrices, which we denote by B_n. The complete flag variety can therefore be written as a homogeneous space GL(n,F) / B_n, which shows in particular that it has dimension n(n−1)/2 over F.

Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the special linear group SL(n,F) of matrices with determinant one, which is a semisimple algebraic group; the set of upper triangular matrices of determinant one is a Borel subgroup.

If the field F is the real or complex numbers we can introduce an inner product on V such that the chosen basis is orthonormal. Any complete flag then splits into a direct sum of one dimensional subspaces by taking orthogonal complements. It follows that the complete flag manifold over the complex numbers is the homogeneous space

where U(n) is the unitary group and Tn is the n-torus of diagonal unitary matrices. There is a similar description over the real numbers with U(n) replaced by the orthogonal group O(n), and Tn by the diagonal orthogonal matrices (which have diagonal entries ±1).