Questions tagged [grassmannian]

In mathematics, the Grassmannian $\mathbf{Gr}(r, V)$ is a space which parameterizes all linear subspaces of a vector space $V$ of given dimension $r$.

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Fundamental Group of complex Grassmannian

How can we compute the fundamental group of complex Grassmannian Gr(k,n) using Van Kampen Theorem only? I know it will be simply connected, so we better find two simply connected open sets which cover ...
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Are dual numbers related to dual spaces? [closed]

I recently came across the notion of a dual number and am curious if there is any direct relation to a dual space. I can't really find anything to answer this, though the notion of a Grassmann number (...
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Given a vector bundle over a compact space, how can we determine the smallest Grassmannian classifying it?

It is a standard result that for $\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$, the Grassmannian $G_n(\mathbb{F}^{\infty})$ is a homotopical classifying space for $n$-plane bundles over any ...
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Lines on grassmannian

Given a projective space $\mathbb{P}^n(\mathbb{C})$, I can consider the Grasmannian of lines $G(2,n+1)$, which has a structure of projective variety inside $\mathbb{P}^N$, where $N=\binom{2+n+1}{2}-1$...
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Compactness of the set of rank K projectors

I hope you could give a hint for proving that the following set is compact $(k<n)$: $X=\left\{A\in \mathbb{R}^{n\times n}:A=A^{t},A^{2}=A,rank(A)=k\right\}$ I can proof that $X$ is bounded(not so ...
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I am dealing with the embedding of real grassmannians $G(n,k)$ on $\mathbb{R}^{n^2}$ via the map associating to each vector space the projection matrix on it in the canonical base of $\mathbb{R}^n$. ...
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Grassmannians: Varieties swept out by linear spaces (Eisenbud & Harris: 3264 and All That)

I have a couple of questions on statements from Harris' and Eisenbuds's lecture "3264 and All That" at page 145, Section 4.2.3: Varieties swept out by linear spaces. The content of 4.2.3 answers ...
Consider the complex projective space $\mathbb{P}^4$ and the Grassmannian $\mathbb{G}(1:\mathbb{P}^4)$ of lines in it, seen as a projective manifold through plucker embedding. Take $l_1,l_2\subset \... 1answer 64 views Is the map assigning$x$to its tangent space smooth? I'm having some difficulties understanding smooth maps between manifolds, and in particular I would like to know if the map$x \mapsto T_{x}M$is a smooth map$M\to Gr(k,n)$? Here$x \in M$, where$M$... 1answer 539 views matrix wise tangent inverse (arctan) Given a matrix$X$, an expression for the matrix cosine and sine are given by $$\textrm{cos}(X) = \frac{e^{iX} + e^{-iX}}{2}\\ \textrm{sin}(X) = \frac{e^{iX} - e^{-iX}}{2i}$$ I have been trying to ... 0answers 74 views Generically transversally intersecting Schubert cycles I have a question about the the proof of Pieri's formula from Harris' and Eisenbuds's lecture "3264 and All That"on page 146. Before the proof we use this terminology (see page 139): let$G=G(k,V)$... 1answer 72 views Is there any entity that possess information of magnitude,direction,starting and ending points? Is there any entity similar to vectors but also possess the starting and ending points ? For instance, consider a plane$z = 4$, Suppose I want a vector starting from A$(0,0,4)$and ending at B$(0,1,...
We have the following Lemma: Lemma Let $\Sigma_a(\mathcal{V}),\Sigma_b(\mathcal{W})$ be two Schubert cycle defined relative to transverse flags $\mathcal{V}$ and $\mathcal{W}$. If \$\Lambda \in \...