Hilbert–Poincaré series: Difference between revisions

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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 space|affine subspaces]] of '''R'''<sup>''n''</sup> (described on this page), while in the other the [[affine Grassmannian]] is a quotient of a group-ring based on formal Laurent series.
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==Formal definition==
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Given a finite-dimensional [[vector space]] ''V'' and a non-negative integer ''k'', then Graff<sub>''k''</sub>(''V'') is the [[topological space]] of all [[Affine space|affine]] ''k''-dimensional subspaces of ''V''.
 
It has a natural projection ''p'':Graff<sub>''k''</sub>(''V'') → Gr<sub>''k''</sub>(''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 <math>p(U)^\perp</math>, 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<sub>''k''</sub>(''V'').
 
As a [[homogeneous space]], the affine Grassmannian of an ''n''-dimensional vector space ''V'' can be identified with
 
:<math> \mathrm{Graff}_k(V) \simeq \frac{E(n)}{E(k)\times O(n-k)} </math>
 
where ''E''(''n'') is the [[Euclidean group]] of '''R'''<sup>''n''</sup> and O(''m'') is the [[orthogonal group]] on '''R'''<sup>''m''</sup>. It follows that the dimension is given by
 
:<math> \dim\left[ \mathrm{Graff}_k(V) \right] = (n-k)(k+1) \, . </math>
(This relation is easier to deduce from the identification of next section, as the difference between the number of coefficients, (''n''&minus;''k'')(''n''+1) and the dimension of the linear group acting on the equations, (''n''&minus;''k'')<sup>2</sup>.)
 
==Relationship with ordinary Grassmannian==
Let {{nowrap|(''x''<sub>1</sub>,&hellip;,''x''<sub>''n''</sub>)}} be the usual linear coordinates on '''R'''<sup>''n''</sup>.  Then '''R'''<sup>''n''</sup> is embedded into '''R'''<sup>''n''+1</sup> as the affine hyperplane ''x''<sub>''n''+1</sup>&nbsp;=&nbsp;1. The ''k''-dimensional affine subspaces of '''R'''<sup>''n''</sup> are in one-to-one correspondence with the linear subspaces of '''R'''<sup>''n''+1</sup> that are in general position with respect to the plane ''x''<sub>''n''+1</sup>&nbsp;=&nbsp;1. Indeed, a ''k''-dimensional affine subspace of '''R'''<sup>''n''</sup> is the locus of solutions of a rank ''n''&nbsp;&minus;&nbsp;''k'' system of affine equations
:<math>
\begin{align}
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{align}
</math>
These determine a rank ''n''&minus;''k'' system of ''linear'' equations on '''R'''<sup>''n''+1</sup>
 
:<math>
\begin{align}
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{align}
</math>
 
whose solution is a (''k''&nbsp;+&nbsp;1)-plane that, when intersected with ''x''<sub>''n''+1</sub>&nbsp;=&nbsp;1, is the original ''k''-plane.
 
Because of this identification, Graff(''k'',''n'') is a [[Zariski open set]] in&nbsp;Gr(''k''&nbsp;+&nbsp;1,&nbsp;''n''&nbsp;+&nbsp;1).
 
==References==
*{{Citation | last1=Klain | first1=Daniel A. | last2=Rota | first2=Gian-Carlo | title=Introduction to Geometric Probability | publisher=[[Cambridge University Press]] | location=Cambridge | year=1997 }}
 
[[Category:Differential geometry]]
[[Category:Projective geometry]]
[[Category:Algebraic homogeneous spaces]]
[[Category:Algebraic geometry]]

Latest revision as of 16:04, 5 January 2015

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