Talk:Affine Grassmannian

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The affine Grassmannian should be defined in a more understandable manner - as the set of k-dimensional affine subspaces of R^n or C^n. Then the more general case of algebraic groups can follow. Simplifix (talk) 21:42, 18 April 2008 (UTC)

If indeed the set of k-dimensional affine subsets of Euclidean space is what it is, then I agree. I seem to recall that in Klain and Rota's book on geometric probability, that's what the term is taken to mean. Michael Hardy (talk) 21:59, 18 April 2008 (UTC)

If that's what this is, then I entreat you, please add this fact. I only wrote what I did because that's the only definition I know (it's how the geometric Langlands people define it). I just needed a place for the link in Beauville–Laszlo theorem to point. Ryan Reich (talk) 00:49, 19 April 2008 (UTC)

Actually I don't know what the relationship is between the finite dimensional affine grassmannian I mentioned above, and the infinite dimensional one that has already been defined. If I write what I proposed, then they would be two different things - and would even perhaps need a disambiguation page! Unless someone else knows a relationship between the two. (Klain and Rota is where I first came across it too but I think it should also be related to the Radon transform) Simplifix (talk) 18:13, 20 April 2008 (UTC)

Are you, perhaps, thinking just of Grassmannian? Because I've never heard of that being called the affine Grassmannian. Or is there another, similarly defined manifold of all k-dimensional hyperplanes in Rⁿ (as opposed to just k-dimensional linear subspaces)? Ryan Reich (talk) 14:49, 23 April 2008 (UTC)

Points in the Grassmannian are linear subspaces; they contain the origin. Points in the affine Grassmannian (as defined in Klain & Rota, if I remember correctly) are affine subspaces; they need not pass through the origin. That's the difference. Michael Hardy (talk) 21:00, 23 April 2008 (UTC)

That's what it sounded like from the description. Isn't then the affine Grassmannian as you define it just an open subspace of the Grassmannian in one higher dimension? Just embed your Rⁿ as the plane x₀ = 1 in R^{n + 1} and intersect linear subspaces with it. Obviously not canonical, though. Ryan Reich (talk) 21:18, 23 April 2008 (UTC)

I've added a new page: Affine Grassmannian (manifold) (for lack of a better name). And yes, it is an open subset of the appropriate Grassmannian, in much the same way as Rⁿ is an open subset of th eappropriate real projective space. Simplifix (talk) 22:32, 23 April 2008 (UTC)

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seems like this page should be combined with the other page on affine grassmannian (manifold) Joshuav (talk) 14:30, 19 June 2010 (UTC)