Generalized Flag Variety - Partial Flag Varieties

Partial Flag Varieties

The partial flag variety

is the space of all flags of signature (d₁, d₂, … d_k) in a vector space V of dimension n = d_k over F. The complete flag variety is the special case that d_i = i for all i. When k=2, this is a Grassmannian of d₁-dimensional subspaces of V.

This is a homogeneous space for the general linear group G of V over F. To be explicit, take V = Fn so that G = GL(n,F). The stabilizer of a flag of nested subspaces V_i of dimension d_i can be taken to be the group of nonsingular block upper triangular matrices, where the dimensions of the blocks are n_i := d_i − d_i−1 (with d₀ = 0).

Restricting to matrices of determinant one, this is a parabolic subgroup P of SL(n,F), and thus the partial flag variety is isomorphic to the homogeneous space SL(n,F)/P.

If F is the real or complex numbers, then an inner product can be used to split any flag into a direct sum, and so the partial flag variety is also isomorphic to the homogeneous space

in the complex case, or

in the real case.