# 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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### Schubert calculus

Let $X = Gr(2,4)$ the complex Grassmannian of $2$-planes in $V = \Bbb C^4$ and $S$ the tautological bundle, $Q$ the quotient bundle. The cohomology ring is generated by $c_1(S), c_2(S)$ with relations ...
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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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### About embeddings of real grassmannians

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 ...
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### Cohomology ring of Gr(2, 5)

Does anyone know how to prove that the grassmannian G(2, 5) is generated by the Schubert cycles or some source where I can find a proof? Thanks!
I'm trying to show that real grassmannians $G(k, n)$ are smooth manifolds of dimension $k(n-k)$. The problem is set in this way: Identify the set of all real matrices with $n$ rows and $k$ columns ...
Let $(R, \mathfrak m)$ be a Noetherian local ring such that the residue field $k=R/\mathfrak m$ is algebraically closed. For a fixed integer $t \ge 1$ , $V_t :=\mathfrak m^t/\mathfrak m^{t+1}$ is a ...