Affine Grassmannian (manifold)

In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all "k"-dimensional affine subspaces of R"n" (described on this page), while in the other the affine Grassmannian is a quotient of a group-ring based on formal Laurent series.
Formal definition.
Given a finite-dimensional vector space "V" and a non-negative integer "k", then Graff"k"("V") is the topological space of all affine "k"-dimensional subspaces of "V".
It has a natural projection "p":Graff"k"("V") → Gr"k"("V"), the Grassmannian of all linear "k"-dimensional subspaces of "V" by defining "p"("U") to be the translation of "U" to a subspace through the origin. This projection is a fibration, and if "V" is given an inner product, the fibre containing "U" can be identified with formula_1, the orthogonal complement to "p"("U").
The fibres are therefore vector spaces, and the projection "p" is a vector bundle over the Grassmannian, which defines the manifold structure on Graff"k"("V").
As a homogeneous space, the affine Grassmannian of an "n"-dimensional vector space "V" can be identified with
where "E"("n") is the Euclidean group of R"n" and O("m") is the orthogonal group on R"m". It follows that the dimension is given by
Relationship with ordinary Grassmannian.
Let be the usual linear coordinates on R"n". Then R"n" is embedded into R"n"+1 as the affine hyperplane "x""n"+1 = 1. The "k"-dimensional affine subspaces of R"n" are in one-to-one correspondence with the ("k"+"1")-dimensional linear subspaces of R"n"+1 that are in general position with respect to the plane "x""n"+1 = 1. Indeed, a "k"-dimensional affine subspace of R"n" is the locus of solutions of a rank "n" − "k" system of affine equations
These determine a rank "n"−"k" system of "linear" equations on R"n"+1
whose solution is a ("k" + 1)-plane that, when intersected with "x""n"+1 = 1, is the original "k"-plane.
Because of this identification, Graff("k","n") is a Zariski open set in Gr("k" + 1, "n" + 1).