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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 , 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 R''n'' and O(''m'') is the orthogonal group on R''m''. It follows that the dimension is given by : (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.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Affine Grassmannian (manifold)」の詳細全文を読む スポンサード リンク
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