# 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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### Group/H-space structures for standard models of classifying spaces?

Let $G$ be a commutative topological group. May in his textbook, A concise course in algebraic topology, gives a model of the classifying space $BG$ so that it is a commutative topological group ...
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### plane as a 4D anti-vector and the distance from the origin

So, we can get a plane in 4D the following way using the wedge product - p ^ q ^ r. It encompasses information on both the normal to the plane and the distance to ...
1 vote
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### Unique representation of $\mathbf{Gr}^+(p,n)$ the oriented real Grassmannian

For $\mathbf{Gr}(p,n)$ the $p$ dimensional subspace of $\mathbb{R}^n$, or equivalently $O(n)/O(p)\times O(n-p)$, a point has a unique projector representation $P = UU’$ where U is an $n \times p$ ...
1 vote
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### Realizing a neighborhood in $\mathbf{Gr}(k,n)$ as a neighborhood in $\mathbf{Gr}(1,n)$

Fix two positive integers $k,n$ with $1 < k < n$ and equip the Grassmannian $\mathbf{Gr}(k,n)$ with your favorite metric $\rho$. Now fix $\delta > 0$ and consider an open ball $B_{\delta}(x)$ ...
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### Chern class of bundle of linear functions on $G(2,4)$

Let $G=G(1,3)$ be the grassmanian of lines in $\mathbb{C}P^3$. Let $E$ be the bundle on $G$ whose fiber over $\ell$ are homogeneous linear functions on $\ell$. I am interested in $c(E)=1+c_1(E)+c_2(E)$...
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### Inverse of Grassmann variables

Let $\theta$ be a Grassmann variable. I know that $1/\theta$ is not defined but $\frac{1}{1-\theta}=1+\theta$. My question is simple: is $\frac{1}{\bar{\theta}\theta}$ defined and if so how? My guess ...
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### Rigorously working with flat limits: lines meeting a curve by specialization

I am trying to get comfortable with flat limits. This question is motivated by Section 3.5.3 of Eisenbud and Harris's '3264 And All That' and Exercises 3.35 and 3.36. This section and the surrounding ...
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### Confusion regarding definition of Grassmannian

I had been thinking of Grass$(p,r)$ as the set of $r$ dim subspaces of $k^p$ where $k$ is whatever ground field we take in the context. We then topologize this set. Now in one of his papers, Nitsure ...
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### Grassmannian as a smooth projective variety

I have to prove that the Grassmannian is a smooth projective variety. I was able to show that it is a projective variety using this script: https://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/...
1 vote
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### Show that $\mathbb{P}(V)$ and $G_{1}(V)$ are diffeomorphic

Be $V$ a real vector space of dimension $n+1$ defined over $\mathbb{R}^{n+1}$. Build up a differential structure on $\mathbb{P}(V)$, the projective space on $V$. Show that there is a diffeomorphism ...
1 vote
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### Deviation of Random Projections

Let $P$ be an orthogonal projection in $\mathbb{R}^n$ onto an $m-$dimensional random subspace uniformly distributed in the Grassmannian $G_{n,m}$. Let $T$ be a bounded subset of $\mathbb{R}^n$. Let $x$...
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### Find the subspace of $\mathbb{C}^n$ with given Plücker coordinates.

I'm studying Representation theory and I'm trying to understand the Grassmanian and Plücker coordinates. This problem is an exercise from Young Tableaux: With Applications to Representation Theory and ...
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### What is the base point of the Stabilizer that forms the subspace $Span${$e_1$,....,$e_k$}

I am preparing for an exam for my linear algebra class and I had this question pop up in my mind. What is the base point for the isotropy group that is the subspace $Span${$e_1$,....,$e_k$} This ...
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### Prove that the group of invertible matrices acts transitively on Grassmannian space

I have recently learnt about group actions and grassmannian space. So While searching further on the topic about general linear groups acting on different sets, I read something on wikipedia: $GL$($V$...
1 vote