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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What is the principal bundle structure of $O(n)$?

Consider the map $\pi:O(n)\rightarrow G(k,n)$ which maps $A\in O(n)$ to the subspace of $\mathtt{R}^n$ spanned by the first $k$ columns of $A$. Here $G(k,n)$ is the Grassmannian manifold. My question ...
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Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
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A question about a universal family over a Grassmannian.

I refer to this paper on Moduli Spaces by Ravi Vakil. I am uploading a screenshot: What can possibly be a universal family over $G(k,n)$? For example, let us take the set of all linear subspaces ...
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Find ideal defining $Gr_2(\mathbb{C}^5)$ in Pluker embedding

Let $Gr_k(\mathbb{C}^n)$ the Grassmannian variety of $k$-planes in the complex space $\mathbb{C}^n$. We can consider the Pluker embedding $$ \mathcal{P}: Gr_k(\mathbb{C}^n) \to \mathbb{P}(\Lambda^k ...
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Finding Grassmann coefficients

If we for example represent the Grassmann variables as $4\times4$ matrices, then is there some procedure to find the matrices $f_i$, so that we will get the following decomposition of some given ...
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Stabilizer of an element in the Grassmannian

Let $Gr_{n,k} = \{W\leq \mathbb{C}^n:\mbox{dim(W)}=k \}$ be the set of $k$ dimensional subspaces of $\mathbb{C}^n$. We have a group action: $$GL_n(\mathbb{C})\times Gr_{n,k} \to Gr_{n,k} \ , \ ...
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Identities for the Hilbert–Schmidt norm of products of projections.

I've been studying different metrics on the Grassmannian $Gr(k,n)$ of k-dimensional linear subspaces of $\mathbb{R}^n$ and found myself needing some identities for the norm of a product of orthogonal ...
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72 views

Is a ruled surface of degree>2 always singular?

Let $X=\mathbb{C}\mathbb{P}^3$ and let $V\subset X$ be a closed algebraic sub variety. By V-is ruled, I mean that for every point in $V$ there is a line passing through it which also lies in V. ...
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Isomorphism between projective space and grassmanian $\mathrm{Grass}(1,n+1)$

I have to show that the projective space $\mathbb{P}^n$ is isomorphic to the grassmanian $\mathrm{Grass}(1,n+1)=\{V\subseteq\mathbb{R}^{n+1}:V\,\text{linear subspace,}\,\dim\,V\,=1\}$ as well as ...
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De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
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Induced measurable subbundle

Let $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ measurable and suppose that the application $$ \displaystyle{\begin{array}{rccl} h:&Z&\longrightarrow& ...
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In search for a topology

I'm looking for a way to convergence on subspaces. If $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ and consider in $G_k(\mathbb{R}^m)$ one topology $\tau$. I would like to ...
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232 views

Convergence equivalence

If $G_k(\mathbb{R}^m)=\{ W: W$ is subspace of $\mathbb{R}^m, \dim W=k \}$ and consider in $G_k(\mathbb{R}^m)$ one topology $\tau$ where $U\in \tau$ is open iff the set $\widehat{U}=\lbrace v: v\in ...
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Is there an embedding of projective varieties $\mathrm{Grass}(r,n)\hookrightarrow(\mathbb{P}^{n-1})^{\times r}$?

Let $k$ be an algebraically closed field, and let $r\le n$ be positive integers. Let $\mathrm{Grass}(r,n)$ be the projective variety of all $r$-dimensional planes in $k^n$. Notice that ...
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40 views

How to understand following expression for gaussian integral of grassmannian function?

Let's have grassmannian numbers $\theta_{i}$: $$ \theta_{i}\theta_{j} = -\theta_{j}\theta_{i}, \quad \theta_{i} x = x\theta_{i}, $$ where x is just ordinary number. How to understand (or how to show ...
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Grassmannian as a submanifold of the exterior product

I'm looking for a proof of the fact that if $V$ is a finitely dimensional vector space, then $G_p(V) \setminus \{0\}$ is a submanifold of $\Lambda_pV$. Here $G_p(V) = \{ v_1 \wedge ... \wedge v_p \ ...
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Isotropy group for the subset of Grassmannian

Consider a complex Grassmannian $Gr_{k}(C^{n})$, which is a symmetric space with symmetry group $U(n)$ (i.e. unitary group). Consider a subspace $S_{0}$ of the Grassmannian determined by the canonical ...
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Grassmannian bundle: any good reference?

I have met the notion of Grassmannian bundle of a vector bundle over a variety in intersection theory, but anywhere I look I just find a brief recall of how the stalks look like (my references so far ...
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41 views

Gaussian for Grassmann variables

Let $(\theta,A\theta)=\theta_i A_{ij}\theta_j$ where $A$ is some $(2\times2)$ antisymmetric matrix. I want to generalize the following $$I(A) =\int d\theta_1d\theta_2~ ...
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146 views

Finding a matrix representation for two Grassmann numbers.

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
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1answer
75 views

Continued matrices-valued function

Given $d<k$. Let ${\cal M}_{d\times k}(\mathbb{R})$ denotes the set of all $d\times k$ real matrices and suppose that $H:\mathbb{R}^k\rightarrow {\cal M}_{d\times k}(\mathbb{R})$ is a continuous ...
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Canonical line bundle over a projective bundle

The following is an excerpt from the Atiyah's K-Theory. If $E$ is a vector bundle over $X$ then each point $a\in P(E)_{x}=P(E_{x})$ represents a one-dimensional subspace $H_{x}^{*}\subset E_{x}$. ...
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Fitting ideals and Grassmannians

Let $L$ be a locally free and finitely presented sheaf over a Noetherian scheme $X$ and $$ E\overset{\varphi}\to F \to L \to 0$$ a free presentation of $L$, where $E$ and $F$ have finite ranks $n$ ...
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Reference on Grassmanian

Can anyone suggest a reference on the Grassmanian which describe the Riemannian structure of the Grassmanian $Gr(k, n)$? Specifically, I want to know about the geodesics, convex neigboorhood, geodesic ...
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Affine stratification of Grassmannian $\mathbb{G}(1,\mathbb{P}^3)$

Let $G=\mathbb{G}(1,\mathbb{P})$ be the Grassmannian variety of lines in $\mathbb{P}^3$. I have to do an affine stratification of $G$. In order to do this we consider the flag $\mathcal{F}$ of the ...
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178 views

Universal quotient bundles of $G(2,4)$ and $\mathbb{G}(1,\mathbb{P}^3)$

Let $V$ be an $n$-dimentional complex vector space, $G=G(k,V)$ the Grassmannian of $k$-planes in $V$, and let $\mathcal{V}:=V \otimes \mathcal{O}_G$ the rank-$n$ trivial vector bundle on $G$. We ...
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101 views

How can I visualize what open sets “look” like in unfamiliar topological spaces?

The question is extremely general, but I do have a specific case I'd like to look at, and I'm hoping that some combination of specific pointers and general advice will help me out. Consider the ...
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Is there relation between Grassmann Manifold and Grassmann Algebra?

I'm an EE student and I'm beginning to learn about the Grassmann Manifold. As is known that the Grassmann Manifold is a space treating each linear subspace with a specific dimension in the vector ...
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why we want to use grassmannian space?

I wonder what's the special about grassmannian space? Why we want to use this space? On wikipedia, it says: "By giving a collection of subspaces of some vector space a topological structure, it is ...
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Non-(stable)-triviality of the tautological bundles

The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold $G_k(\mathbb{K}^N)$ of all $k$-planes in $\mathbb{K}^N$ (for $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$ or ...
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Derivation of Grassmann valued functional

I'm trying to evaluate $$\frac{\delta}{\delta \eta(x)}e^{-\int dz \theta^*(z)\eta(z)}$$ Where $\theta^*(x)$ and $\eta(x)$ are Grassmann valued functions. The context of the functional is in term of ...
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Why is Grassmanian a projective variety? [duplicate]

The grassmanian $ \mathbf G(r,n)$ is the set of all $k$-dimensinal subspaces of a $n$-dimensional vector space. I understand how $ \mathbf G(r,n)$ can be embebbed in the projective space $\mathbb ...
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Symmetry of a Plücker function

Let $d \in \mathbb{N}$ and let $I$ be a set. Let $\omega : I^d \times I^d \to \mathbb{R}$ be a function, denoted by $(a_1,\dotsc,a_d,b_1,\dotsc,b_d) \mapsto a_1 \cdots a_d | b_1 \cdots b_d$, with the ...
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The dimension of linear map

I am reading "Introduction to smooth manifolds" by Lee and one place is very unclear for me: Let $P$ and $Q$ be any complementary subspaces of $V$ (which is an $n$-dimensional real vector space) of ...
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intrinsic proof that the grassmannian is a manifold

I was trying to prove that the grassmannian is a manifold without picking bases, is that possible? Here's what I've got, let's start from projective space. Take $V$ a vector space of dimension n, and ...
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Grassmann Variables and Complex Conjugate

While dealing with Grassmann Variables, the complex conjugate is defined as $$ (\phi \psi)^{\dagger} = \psi^{\dagger} \phi^\dagger $$ and why not $ \phi^{\dagger} \psi^\dagger $. I want to know the ...
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Openness of $\varphi(U_Q \cap U_{Q'})$ in the definition of Grassmannian Manifolds (Lee: Introduction to Smooth Manifolds)

I am reading Lee's Introduction to Smooth Manifolds and I have some problems with definition of Grassmannian manifold given in Example 1.24, p.22. I'll write the details below. My question is: Why ...
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Different definitions of the complex Grassmannian

I have come across two different definitions of what I suspect is the same object. Both are called the complex Grassmannian: 1: $U(n)/U(k)\times U(n-k)$ 2: $SU(n)/(S(U(k)\times U(n-k))$ What is the ...
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Finding the singularities of affine and projective varieties

I'm having trouble calculating singularities of varieties: when my professor covered it it was the last lecture of the course and a shade rushed. I'm not sure if the definition I've been given is ...
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243 views

Writing down cohomology groups of the complex Grassmannian

I am studying a homogeneous space and would like to know its cohomology groups. Using some sequences and fibrations I have figured out some of these groups, but largely in terms of the cohomology ...
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Tautological vector bundle over $G_1(\mathbb{R^2})$ isomorphic to the Möbius bundle

Let $V$ be a finite dimensional vector space, and let $G_k(V)$ be the Grassmannian of $k$-dimensional subspaces of $V$. Let $T$ be the disjoint union of all these $k$-dimensional subspaces and ...
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How does a Riemannian metric “naturally induce” distances in the Grassmannian bundle

When reading about Pesin theory I've run into the necessity of defining a metric on the Grassmannian bundle of a compact Riemannian manifold $M$. More specifically a fiber at $x \in M$ in the ...
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Fundamental groups of Grassmann and Stiefel manifolds

Could someone provide details on how to compute fundamental groups of real and complex Grassmann and Stiefel manifolds?
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What are the complex points of the real Grassmann variety?

If $X=Gr_{n,k}(\mathbb{R})$ is a real Grassmann variety (of $k$-planes in $n$-dimensional space), then what is $X(\mathbb{C})$, the set of complex points of $X$? In particular, can it be identified as ...
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Generalization of Plücker embedding

Let $V$ be a vector space and $1 \leq k \leq n$ natural numbers. By $\operatorname{Grass}_n(V)$ I mean the Grassmannian of $n$-co​dimensional subspaces of $V$, that is, $n$-dimensional ...
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When does variété mean manifold?

Following advice from this post, I am in the process of translating Ehresmann's 1934 paper "Sur la Topologie de Certains Espaces Homogènes" from French to English. French-English dictionaries online ...
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Topology on the general linear group of a topological vector space

Let $K$ be a topological field. Let $V$ be a topological vector space over $K$ (if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of ...
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What are the attaching maps for the real Grassmannian?

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. The study of characteristic classes tells us that these ...
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How to find a Zariski Cover of the Grassmannian

I am wondering how to find the Zariski Cover of the Grassmannian over $\mathbb{C}$. I was hoping some reference would go through this calculation. I was recently told that it was not too hard, so I ...