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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Correspondence defines embedding of $G_n(\mathbb{R}^m)$ into $G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = G_{n+1}(\mathbb{R}^{m+1})$? [closed]

Does the correspondence$$X \overset{f}{\to} \mathbb{R}^1 \oplus X$$defines an embedding of the Grassmann manifold $G_n(\mathbb{R}^m)$ into $$G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = ...
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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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Trying to understand some basic facts about tangent space of Grassmannian.

I am reading Harris's 'Algebraic Geometry: A first course'. I am trying to understand its identification of the tangent space to a Grassmannian. Let $G(k,n)$ be the Grassmannian of $k$-planes in ...
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The oriented Grassmannian $\widetilde{\text{Gr}}(k,\mathbb{R}^n)$ is simply connected for $n>2$

I saw this result mentioned a lot in many references, but it is always stated as a fact or an exercise. My approach would be to see the oriented grassmannian as the quotient ...
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Decomposition into simple bivectors

According to Wikipedia, any element of $\wedge^2\Bbb R^n$ should be decomposable into $n/2$ simple bivectors for $n$ even or $(n-1)/2$ for $n$ odd. How do I count that? How do I check that ...
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59 views

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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Generalized Gauss map, giving rise to second fundamental form

I know that the tangent bundle of $G_n(\mathbb{R}^{n+k})$ is isomorphic to $\text{Hom}(\gamma^n(\mathbb{R}^{n+k}), \gamma^\perp)$, where $\gamma^\perp$ denotes the orthogonal complement of ...
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Direct Limit of Grassmannians

Let $X$ be a topological space and $G_n(\mathbb{C}^m)$ be the space of vector subspaces of $\mathbb{C}^m$ of codimension $n$. Let $G_n(\mathbb{C}^\infty):=\bigcup_{m=n}^{\infty}G_n(\mathbb{C}^m)$ ...
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Full flag $Fl_{\mathbb C}(3)$

How we can see that the full complex flag when $n=3$ is equivalent to one of these spaces: $\{(u,v)\in \mathbb CP^2\times \mathbb CP^2 ; u\perp v\}$ and what is dimension over $\mathbb C$ here? ...
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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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55 views

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 ...
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Why is the dimension of this Grassmann manifold $G_{d, n}$ equal to $(d+1)(n-d)$ formed by the Plucker coordinates of a $d$-plane?

A $d$-plane $L \subset \mathbb{P}^{n}$ is defined as the set of points $P=(p(0), p(1), \ldots, p(n)) \in \mathbb{P}^{n}$ that satisfy equations $\sum_{j=0}^{n} b_{\alpha j}p(j) = 0$, where $\alpha = ...
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Is There a Generalization of the Path Lifting Property Of Covering Maps.

$\newcommand{\R}{\mathbf R}$ Let $p:(E, e)\to (X, x)$ be a covering projection map. We know that for any path $\gamma:I\to X$ such that $\gamma(0)=x$, there is a unique lift $\Gamma:I\to E$ such that ...
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A Proof of the Hausdorffness of the Grassmannian Using the Basics

$\DeclareMathOperator{\Span}{span} \newcommand{\R}{\mathbf R} \newcommand{\mc}{\mathcal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\grassman}{GR} \newcommand{\set}[1]{\{#1\}} ...
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Schubert calculus on Grassmannians

Can anyone please suggest me some notes or books where I can read about Schubert calculus? I am studying Grassmannian varieties so I would like to understand how to use this tool, in particular with ...
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49 views

Homology groups of a Grassmannian

I can't manage to calculate the homology and comology groups of the Grassmannian $G(k+1,n+1)$. I know the definition of homology group and I read which are the homology groups of a Grassmannian, but I ...
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Closure of Schubert cell is the Schubert variety

My question concerns Proposition 1.4.6 in the following article: http://www.mi.uni-koeln.de/~littelma/SMTkurz.pdf . There's just one, apparently straightforward detail of the argument which I can't ...
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Integral homology of real Grassmannian $G(2,4)$

I would like to compute $\pi_1$ and the integral homology groups of the real Grassmannian $G(2,4)$. (This is a question on an old qualifying exam.) The hint for the computation of $\pi_1$ is to put a ...
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Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
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Hermitian metric on line bundle over the Grassmannian

We know that the Grassmannian manifold $G(k,\mathbb{C}^n)$ can be embedded in the projective space $\mathbb{C}P^N$ for $N= {n\choose k}-1$ by the Plucker embedding $P$. On $\mathbb{C}P^N$ we have ...
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Classification of rank $\geq 2$vector bundles over Grassmanians

Are there classification results of higher rank (complex) vector bundles over (complex) Grassmanian manifolds? For example, we know that line bundles are in correspondence with the $H^2(G)$, the ...
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What is the dimension of the space of planes in $\Bbb R^3$?

What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer? Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...
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How does the multiplicative group of a finite field, considered as a vector space, act on subspaces?

Given that a finite vector space $V = \operatorname{GF}(p)^n$ corresponds to the finite field $F = \operatorname{GF}(p^n)$, I'm wondering about how the multiplicative subgroup of $F$, $F^*$, acts on ...
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Complexity of finding M nodes in a graph to maximize the pairwise minimum distance between nodes

I want to know the complexity of finding a set of M nodes, $\{U_1,\dots,U_M\}$, in a given graph $G$, to maximize $d(U_i,U_j)$ over all pairs $i\neq j$, where $d(\cdot,\cdot)$ is the length of the ...
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The last coordinates of basis vectors are a chart: mistake in this example?

While trying to understand local chart on Grassmannians I came across this example in this book: Take $V = \mathbb R^2$ and $U,W$ two subspaces generated by linearly independent vectors. The books ...
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The last coordinates of basis vectors are a chart

Let $Gr$ denote the Grassmannian and let $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} Gr(2, T_x \mathbb R^3)$. Consider one $2$-dimensional subspace of $\mathbb R^3$, that is, one element of ...
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Question about vectors in the Grassmannian in this example

Consider $f: \mathbb R^2 \to \mathbb R^3$ defined by $(t,s) \mapsto (t^2 + 2s, t^3 + 3ts, t^4 + 4t^2 s)$. Let $Gr$ denote the Grassmannian and let $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} ...
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Dimensions of Grassmannians?

I'm trying to work out the dimensions of some examples of Grassmannians but I can't seem to do it. Here is what I understand: The Grassmanian $G(k,n)$ is the set of all $k$ planes in $\mathbb ...
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Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff

$\newcommand{\R}{\mathbf R}$ Let $V$ be an $n$-dimensional vector space and $k$ be an integer less than $n$. A $k$-frame in $V$ is an injective linear map $T:\R^k\to V$. Let the set of all the ...
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A Continuous Choice of $k$-Subspaces of a Vector Space Gives a Continuous Choice of Bases

$\newcommand{\R}{\mathbf R}$ The Grassmannian $G_k(\R^n)$ as a topoplogical space is defined in the following way: Let $F_k(\R^n)$ be the collection of all the linearly independent lists of size $k$ ...
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Proving that certain incidence correspondence is a projective variety.

Let $M$ be the projective space of nonzero $m\times n$ matrices up to scalars (in $\mathbb{K}$). In Joe Harris' Algebraic Geometry: A first course, in order to find the dimension of $M_{k}=\{A\in ...
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Inclusionwise maximal linear subvarieties of a projective variety

Let $X\subseteq\mathbb P^n$ be a complex, projective variety. A linear subspace $L\subseteq\mathbb P^n$ will be called a maximal linear subspace of $X$ if $L\subseteq X$ and for any linear subspace ...
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Diameter of the Grassmannian

Just an interesting question that came to my mind while studying(!): Since the Grassmannian $G(k,\mathbb{C}^n)$ is a compact manifold, what do we know about its diameter? Do we know any estimate? ...
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Motivation for Grassmannian variety

I need some information about the Grassmanian variety for my final project in algebraic geometry course that I am taking. My questions are: Why do we define the Grassmannian variety? Do we use ...
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143 views

Ricci curvature of the Grassmannian?

Let $G(k, \mathbb{C}^n)$ be the Grassmannian of $k-$dimensional complex linear subspaces of $\mathbb{C}^n.$ We know that the Grassmannian can be embedded to the projective space ...
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When is a wedge decomposible?

Shaferevich (in AG book 1) tells me that a wedge $x \in \bigwedge ^r V$ is decompsible iff $(y . x) \wedge x = 0$ for all $y \in \bigwedge^{r-1} V$, where $.$ is the convolution pairing. I see in ...
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Rules for integrating a Grassmann quantity

So I've read that for Grassmann numbers, integration is the same as differentiation. Indeed, on Wikipedia, integration of Grassmann quantities are defined so this is true: $$\int 1 \, d \theta = 0$$ ...
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Universal Property of the Universal Line

In "An Invitation to Quantum Cohomology" by Kock and Vainsencher, they talk about "the universal line", which is defined as the variety $U=\{ (L,p)\in Gr(1,\mathbb{P}^r)\times \mathbb{P}^r | p\in L ...
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Grassmannian a Fano manifold?

I am interested in answering the following: Is it true that the Grassmannian $G(k,\mathbb{C}^n)$ is a Fano manifold? How can I see if its anticanonical bundle is ample?
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Is the determinant bundle the pullback of the $\mathcal O(1)$ on $\mathbb P^n$ under the Plucker embedding?

Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plucker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = ...
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Is it possible to continuously choose one-dimensional subspace in each k-dimensional subspace?

Does there exist a continuous map from Grassmann manifold to projective space $Gr^n(V) \to \mathbb P(V)$, such that image of every n-dimensional subspace lies (1-dimensional subspace) in this ...
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Path-connectedness of Grassmannian

Let $\mathrm{Gr}(k,X^n)$ be Grassmannian. How to prove that $\mathrm{Gr}(k,X^n)$ is path-connected (without using homology theorem) ?
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How to think Grassmannian as a projective variety?

I'm just looking for some explanation for the grassmannian as a projective variety and pl├╝cker embedding.
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Corresponding a vector subspace to a point of the space.

In this article http://en.wikipedia.org/wiki/Tautological_bundle we read: Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector ...
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A question on Grassmannian manifolds and universal line bundles

Problem. Fix an $ n \in \mathbb{N} $. Is it true that there exists a $ k \in \mathbb{N} $ such that every smooth complex line bundle over a smooth $ n $-dimensional real manifold is the pullback of ...
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Hypermatrices, hyperdeterminants and Grassmannians.

Let $Gr(k,n)$ the Grassmannian manifold of the $k$-planes in $\mathbb{C}^n$ and consider the Plucker embedding $\pi: Gr(k,n) \to \mathbb{P}(\Lambda^k \mathbb{C}^n)$. Let $A$ be the set of $n \times n$ ...
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Is the Cone over Grassmannian manifold a determinantal variety?

Let consider the Grassmann manifold $Gr(k,n)$ in the Plucker embedding and the Cone over $Gr(k,n)$, say $C(Gr(k,n))$. On the other hand consider $M$ the set of $n \times n$ skew-symmetric matrices. ...