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.

[edit] 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 , 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 Rⁿ.

[edit] References

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

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