# Tagged Questions

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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### Global sections of dual of the universal bundle on the Grassmanian

Let $G=G(k, V)$ be the Grassmanian of $k$-dimensional subspaces of the $n$th dimensional vector space $V$, regarded as a smooth algebraic variety over $\mathbb{C}$. Denote with $S$ the tautological (...
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### Isomorphism between homotopy groups of Lie group, Grassmann manifold

It is asserted without proof in a book edited by Novikov and Rokhlin that $$\pi_{k - 1}(\text{GL}_n^+(\mathbb{R})) \cong \pi_k(\tilde{G}_n).$$ I know how to show that these two spaces are bijective. ...
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### How to show that the Grassmannian is irreducible as a variety?

How to show that the Grassmannian is irreducible as a variety? I have no idea.
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### Does every element in exterior algebra have the form $v_1 \wedge v_2 \wedge … \wedge v_k$? [duplicate]

Does every element in $\wedge ^k V$ can be expressed as the form $v_1 \wedge v_2 \wedge ... \wedge v_k$ ? Here $V$ is a n-dim vector space, and $v_i$ are vectors in $V$. Intuitively it is right, but ...
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### Quantifying the angle metric on the Grassmannian in terms of the norm on the exterior power

Let $V$ be a finite-dimensional Hilbert space and $G_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$. The $k$th exterior power $\bigwedge^k(V)$ can be equipped with a scalar product by ...
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### How do I find a smooth map from complex Gr(k, n) to real Gr(2k, 2n)?

I am trying to find a smooth bijective map from complex Grassmannian of $k$-dimensional subspaces of $\mathbb{C}^n$ to the Grassmannian $2k$-dimensional subspaces of $\mathbb{R}^{2n}$, but I do not ...
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### What does “linear and injective on each fiber” really mean?

The question is about the proof of the following result: For a paracompact space $B$, the map $[B, \operatorname{Gr}_k] \to \operatorname{Vect}^k(B)$, $[f] \mapsto f^*(\gamma_k)$ is a bijection, ...
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### Can the cohomology of the Grassmannian identified with the cohomology of a specific dense open subvariety?

Let $(\mathbb{C}^{2p},Q)$ be a $2p$-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form $Q$ where $p\geq 3$. Let $l\leq p-2$. You may assume that $l$ is odd if this ...
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### Schubert decomposition of a Grassmannian

I'm going through Sheldon Katz's Enumerative Geometry and String Theory, and a few things regarding the Grassmannian $G(2,4)$ (lines in projective space) are bothering me: How can I compute the ...
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### Understanding the differentiable structures on Grassmann manifold

I am reading in the book Differential Analysis on Complex Manifolds by Raymond. I have a trouble in understanding the differentiable structures on Grassamann space. I uploaded the picture of the page ...
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### Construction of Grassmann manifolds

Is there a way to construct the Grassmann manifold via block matrices? For example the upper triangular matrices stabilize the (coordinate) basis of $\mathbb R^n$.
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### Subvariety of grassmannian of lines

I was trying to prove the following thing, which I think is true but don't see how to handle it... Let $X\subset\mathbb{P}^N$ a projective variety, and let $\mathbb{G}(1,N)$ the Grassmannian of lines ...
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### Isoclinic rotations in four dimensions

Given any collection of complementary, oriented (2D) planes in n-dimensional space, and an angle associated with each one, there is a unique rotation of the whole space which restricts to rotations in ...