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 $p(U)^perp$, 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

:$mathrm\{Graff\}\_k(V)\; simeq\; frac\{E(n)\}\{E(k)\; imes\; O(n-k)\}$

where "E"("n") is the Euclidean group of R^"n".

References

* Daniel A. Klain, Gian-Carlo Rota, Introduction to Geometric Probability, Cambridge University Press, Cambridge, 1997.