# 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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### Grassman variable and Grassmannian?

Grassman variables are anticommuting number or supernumber, is an element of the exterior algebra over the complex numbers. Grassmannian $Gr(k, V)$ is a space that parameterizes all $k$-dimensional ...
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### Grassmannian is a manifold in a specific case the $2$-planes in $\mathbb{R}^4$

I want to show that Grassmannian is a manifold in a specific case the $2$-planes in $\mathbb{R}^4$. I'm in the following context: $G(2,4)$ are the $2$-planes in $\mathbb{R}^4$ that we can identify ...
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### Lagrangian grassmannian of all lagrangian subspaces in $\mathbb{R}^n\times \mathbb{R}^n$ can be identified with $U(n,\mathbb{C})/O(n,\mathbb{R})$

I've been trying to proof this using $U(n,\mathbb{C})$ action over all lagrangian subspaces of $\mathbb{R}^n\times \mathbb{R}^n$ but it didn't work. I mean, I got stuck and I didn't know what else to ...
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### Grassmanian is a manifold (Hirsch)

I am trying to do an exercise from Hirsch's Book , Differential Topology, that basically wants me to prove that $G_{n,k}$ is a manifold. The Grassmanian manifold $G_{n,k}$ of $k$-dimensional ...
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### Ring of functions on the Grassmannian

Background: Let $k$ be a commutative ring. In general for a $k$-functor $X : \textbf{Alg}_k \to \textbf{Set}$ we define the ring of functions on $X$ to be $\textbf{Nat}(X , \mathbb{A}^1)$ with the ...
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### Show that the $GL(n,\mathbb R)/P_k$ is isomorphic to the $GL(n,\mathbb R)$-set grassmannian.

Let $r<n$ be two positive integers and $G=GL(n,\mathbb{R}).$ If $Gr(k,\mathbb{R}^n)$ is the set of all $k$-subspaces, then show that the $G$-sets $Gr(k,\mathbb{R}^n)$ and $G/P_k$ is isomorphic, ...
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### Why are rational functions invariant under symmetric linear group generated by Plucker coordinates

I am working on an exercise and not sure where to start. Let $K=(\mathbb{C}^2)^{n+3}$. The special linear group $SL_2$ acts naturally on each $\mathbb{C}^2$ and hence on $K$. Let $R$ be the field of ...
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### Are linear subspaces of dimension $d$ that exclude a variety of dimension $n-d-1$ an open subset of the Grassmannian?

Suppose I'm working in $\mathbb{P}^n$ and I have an irreducible algebraic variety $X$ of dimension $n-d-1$. In the Grassmannian of dimension $d$, can I always find an open set $U$ such that none of ...
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### Oriented Grassmann is a $2$-sheeted covering space of Grassmann

Let $G_n(\Bbb R^k)$ denote the Grassmann manifold (consisting of all $n$-planes in $\Bbb R^k$), and let $\tilde{G}_n(\Bbb R^k)$ denote the oriented Grassmann manifold, consisting of all oriented $n$-...
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### 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 $M$ ...
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### 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 $G=G(k,V)$...
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### Basis theorem for the Grassamannian [duplicate]

Does anyone know where I can find a good proof for the basis theorem of the cohomology ring of the Grassmannian, or give me a sketch of the proof? I'm already familiar with basic Schubert Calculus.
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### which mehod can easily design index of Grassmannian and its k and n or any function for this in Macaulay2 and how to convert poset to this index?

how to know k and n and its index of Grassmannian? which mehod can easily design index of Grassmannian and its k and n or any library or function for this in Macaulay2? is there any library or ...
The following is from Roman Vershynin's: High-Dimensional Probability: An Introduction with Applications in Data Science. Let $P$ be the orthogonal projection in $\mathbb{R}^n$ onto an $m-$...