Algebraic homogeneous spaces

In mathematics, a Grassmannian \mathbf{Gr}_k(V), also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V over a field K that has a differentiable structure. For example, the Grassmannian \mathbf{Gr}_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf{P}(V) of one dimension lower than V. When V is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsin

homogeneous space which is the quotient of a semisimple group by its parabolic subgroup