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 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 ${\displaystyle p(U)^{\perp }}$, 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 Graff_k(V).

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

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

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

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

(This relation is easier to deduce from the identification of next section, as the difference between the number of coefficients, (n−k)(n+1) and the dimension of the linear group acting on the equations, (n−k)².)

Relationship with ordinary Grassmannian

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

${\displaystyle {\begin{aligned}a_{11}x_{1}+\cdots +a_{1n}x_{n}&=a_{1,n+1}\\&\vdots &\\a_{n-k,1}x_{1}+\cdots +a_{n-k,n}x_{n}&=a_{n-k,n+1}.\end{aligned}}}$

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

${\displaystyle {\begin{aligned}a_{11}x_{1}+\cdots +a_{1n}x_{n}&=a_{1,n+1}x_{n+1}\\&\vdots &\\a_{n-k,1}x_{1}+\cdots +a_{n-k,n}x_{n}&=a_{n-k,n+1}x_{n+1}.\end{aligned}}}$

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).

References

• Klain, Daniel A.; Rota, Gian-Carlo (1997), Introduction to Geometric Probability, Cambridge: Cambridge University Press