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 Rn (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 Graffk(V) is the topological space of all affine k-dimensional subspaces of V.

It has a natural projection p:Graffk(V) → Grk(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 p(U)^\perp, 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 Graffk(V).

As a homogeneous space, the affine Grassmannian of an n-dimensional vector space V can be identified with

\mathrm{Graff}_k(V) \simeq \frac{E(n)}{E(k)\times O(n-k)}

where E(n) is the Euclidean group of Rn and O(m) is the orthogonal group on Rm. It follows that the dimension is given by

\dim\left[ \mathrm{Graff}_k(V) \right] = (n-k)(k-1) \, .

Relationship with ordinary Grassmannian

Let (x1,…,xn) be the usual linear coordinates on Rn. Then Rn is embedded into Rn+1 as the affine hyperplane xn+1 = 1. The k-dimensional affine subspaces of Rn are in one-to-one correspondence with the linear subspaces of Rn+1 that are in general position with respect to the plane xn+1 = 1. Indeed, a k-dimensional affine subspace of Rn is the locus of solutions of a rank n k system of affine equations

\begin{align} a_{11}x_1 %2B \cdots %2B a_{1n}x_n &=& a_{1,n%2B1}\\ &\vdots&\\ a_{n-k,1}x_1 %2B \cdots %2B a_{n-k,n}x_n &=& a_{n-k,n%2B1}. \end{align}

These determine a rank n−k system of linear equations on Rn+1

\begin{align} a_{11}x_1 %2B \cdots %2B a_{1n}x_n &=& a_{1,n%2B1}x_{n%2B1}\\ &\vdots&\\ a_{n-k,1}x_1 %2B \cdots %2B a_{n-k,n}x_n &=& a_{n-k,n%2B1}x_{n%2B1}. \end{align}

whose solution is a (k+1)-plane that, when intersected with xn+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).

References