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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Questions on grassmannians from Harris' and Griffith's principles of algebraic geometry

I have a couple of questions about the explanations on the Grassmannian from Harris' and Griffith's "Principles of algebraic geometry" on page 194/195. We consider for $k \le n$ the $k$-th ...
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$C^m\to C^{m-1}$ projection map projecting out last factor inducing $G_n(C^{m-1})\to G_n(C^m)$?

Assume $m>n$. Consider projection map $\pi: C^m\to C^{m-1}$ by $(z_1,\dots, z_m)\to (z_1,\dots, z_{m-1})$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$ and endow the topology as a ...
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Let $h,h'$ be hermitian metric over vector space $V$, then grassmanian $G_n(V_h)\to G_n(V_{h'})$ is always continuous?

Let $V$ be a finite dimensional vector spaces over complex number and choose hermitian metrics $h,h'$ over $V$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$. Since $V$ has 2 metrics, ...
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Grassmannian is homogeneous, isotropic, and symmetric

I'm trying to prove the Grassmann manifold $\mathrm G_k(\mathbb R^n)$ of $k$-dimensional linear subspaces of $\mathbb R^n$ is isotropic and symmetric. By "isotropic" I specifically that for every ...
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Is the Grassmannian a unique algebraic variety?

I'm fairly new to algebraic geometry, so this is may just be fairly simple question about when two varieties are "the same" variety. In this question, I noted that the Grassmannian can be expressed ...
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What is the way to write a manifold of manifolds?

I want to write this in mathematical notation: "Let us represent a ball, $B_3$, with a metric $g$ as a point on manifold. Let $M$ be the (infinite dimensional) manifold formed from every ball with all ...
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What, exactly, is Schubert's symbolic calculus?

Most modern treatments of the Schubert calculus typically write about the cohomology ring of the Grassmannian. They also write, almost as an afterthought, that this is derived from Schubert's "...
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Stratify Grassmannian based on dimension of intersection

Consider a vector space $V$ of dimension $n$ and an automorphism $A:V\rightarrow V$. Using this we can define, $\Sigma_i = \{W\in Gr_k V| \dim(W\cap AW) = i\}$ for $i=0,\ldots, k$. My question is ...
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The “true” metric on the Grassmannian: Plücker vs Projective-Frobenius embeddings

There are several references in the literature to some kind of "most natural" metric on the Grassmannian manifold, often called the "geodesic distance" or the "Binet-Cauchy" distance. There are ...
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Grassmannian as a quotient of orthogonal or general linear group

I'm trying to understand some different ways to construct the Grassmannian of a real vector space, but I'm having trouble getting some of the notation and definitions. One definition that I often see ...
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Minimal embedding of the Grassmannian into Euclidean (or projective) space

Let $Grass(r,k)$ be the set of all $r$-dimensional subspaces of $\Bbb R^k$. It is well known that $Grass(r,k)$ embeds isometrically as a projective variety into the projectivization of the r'th power ...
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Ring structure of $\Bbb CP^n$ and Chern class.

In this notes Prop 1.71 in nlab, the author aims to compute $H^*(\Bbb C P^n, \Bbb Z)$. I have two confusions. What makes it justified to use $c_1$ as the generator of $H^2(\Bbb CP^n, \Bbb Z)$? There ...
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Does the orthogonal complement determine the inner product up to scaling?

Let $V$ be a real $n$-dimensional vector space, and let $g,h$ be two inner products on $V$. Fix some $1\le k\le n-1$, and denote by $\text{Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$...
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Intersection of lines in $ \mathbb{P}^3 $ can be given as zero locus of homogeneous linear polynomial in $ \mathbb{P}^5 $

I'm currently stuck at Exercise 8.19 b) in the notes to Algebraic Geometry from Gathmann Let $L \subset \mathbb{P}^3$ be an arbitrary line. Show that the set of lines in $\mathbb{P}^3$ that ...
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Meaning of stable $CP^2$

I came across the following phrase in arXiv:1903.08904 ....in order to have a stable $CP^2$ , i.e., one in which all the automorphism group is fixed... Can anyone explain to me what one means by ...
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Does the action of a linear map on $k$-dimensional subspaces determine it up to scaling?

Let $V$ be a real $d$-dimensional vector space, and let $1 \le k \le d-1$ be a fixed integer. Let $A,B \in \text{Hom}(V,V)$, and suppose that $AW=BW$ for every $k$-dimensional subspace $W \le V$. Is ...
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Which subspaces of exterior power have decomposable bases?

Let $V$ be a real $n$-dimensional vector space, and let $1<k<n,r>1$. I wonder: Is there a way to characterise which $r$-dimensional subspaces of the exterior power $\bigwedge^k V$ have ...
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Grassmannians as Gelfand Pairs

Why are $(O(n), O(k) \times O(n-k))$ and $(U(n), U(k) \times U(n-k))$ (corresponding to the real and complex Grassmann manifolds) symmetric Gelfand Pairs? Is this true for $(Sp(n), Sp(k) \times Sp(n-...
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Is there a natural way to view $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d$ as a subspace of $\bigwedge^k_{\mathbb{R}}\mathbb{C}^d$

Does there exists an $\mathbb R$-linear embedding $\bigwedge^k_{\mathbb{C}}\mathbb{C}^d \to \bigwedge^k_{\mathbb{R}}\mathbb{C}^d$ which maps decomposable tensors to decomposable tensors? (The ...
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Is “being decomposable” preserved under taking a subspace?

Let $V$ be a vector space over some field, and $W \le V$ a vector subspace. Let $1<k<\dim V$ be an integer. Suppose $\omega \in \bigwedge^k W$ is decomposable as an element in $\bigwedge^k V$....
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If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ is a complex power, is it a real power up to a sign?

Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$. Does there exist $M \in \...
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Projecting a projective variety away from a linear subspace

I read in Harris' book at page 148, proposition 11.37 and got slightly confused regarding the argument. Harris mentions a projective space $\mathbb{P}^{2n+1}$ and a linear subspace $\mathbb{P}^{n}$ ...
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Drawing random subspaces from Grassmannian with uniform probability

Consider the Grassmannian manifold $G(M, N)$ of $M$-dimensional subspaces in $R^N$. I want to approximate (stochastically) an integral of the form $$ \int_{G(M, N)} f(v) \, dv, $$ where $f : G(M, N) \...
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Isotropy/little group of $O(n)$

I'm trying to prove that the little group of $O(n)$ acting on a $k$-dimensional subspace of $\mathbb{R}^n$, call it $V$, is $O(k)\times O(n - k)$ due to the Grassmann manifold is isomorphic to $O(n)/(...
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Real Grassmann manifold and orthonormal groups

I'm trying to prove that the Grassmann manifold $$G_k(\mathbb{R}^n) = \{E = {\rm {\it k} - dimensional\ subspace\ of\ } \mathbb{R}^n\}$$ is equivalent to: $$G_k(\mathbb{R}^n) = \frac{O(n)}{O(k)\...
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Straight things on Grassmanians

I am new to Grassmanian, so this question may be too easy. But I didn't find it in any books I know. When we talk about Grassmanian, it should not be only a manifold or a variety. At least I think we ...
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General statement about how many lines in Euclidean space will determine a line

It is easy to see that in $3$-dimensional Euclidean space, given $4$ lines in general position, there exists precisely one line who intersects with each of the $4$ lines. We call the $4$ lines ...
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Compute the degree of a particular morphism

Let $\mathbb P^n=\mathbb{CP^n}$ be complex projective space. Let $H^0(\mathcal O_{\mathbb P^n}(d))$ be the group of homogeneous polynomials of degree $d$, and denote its dimension by $N(d)$. Consider ...
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Geometric intuition of the dimension of Grassmannians and flag manfolds [duplicate]

I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{...
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Jet prolongation of a distribution on a manifold

I'm trying to work with the first jet prolongation of a $k$-distribution on a manifold $M$ of dimension $n$. My intuition is to consider the Grassmann bundle $X=Gr_k(TM)\to M$ and look at the first ...
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The Grassmannian has a non-orientable tangent bundle (in a certain sense)

Let $X$ be a smooth scheme of dimension $r$. Given a rank $r$ vector bundle $\pi: E\to X$, we say that $E$ is orientable if there is a line bundle $L$ on $X$ with an isomorphism $L^{\otimes 2} \cong \...
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Relating pullbacks of tautological bundles

Let $X$ be a projective variety and let $\mathcal{E}$ be a vector bundle of rank $r$ on $X$ which is generated by its global sections $V=\Gamma(X,\mathcal{E})$. Recall that this gives us a map $f:X\to\...
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How to optimize objective in the Grassmann manifold?

For Stiefel manifold, it contains all the orthogonal column matrices $$St(d,M) = \{X \in R^{M \times d} | X^TX = I\}$$ For Grassmann manifold, it is $$Gr(d,M) = \{col(X), X \in R^{M \times d}\}$$ ...
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How to optimization the $f(X)= \| A - XX^T \|_F^2 + \| X \|_F^2 $ on Grassmann manifold

$$\begin{array}{ll} \text{minimize} & f(X)= \| A - XX^T \|_F^2 + \| X \|_F^2\\ \text{subject to} & X \in Gr(d,N) \end{array}$$ where $Gr(d,N)$ means the Grassmann Manifold; $\|\cdot\|_F$ ...
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Every Schubert cycle a Chern class?

Consider the Grassmann variety $\mathbb{G}(k,n)$ and its Chow ring $A$. It is known that the classes of Schubert cycles form a $\mathbb{Z}$ basis of $A$. Is it known which of these Schubert cycles can ...
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Schubert class in the Grassmannian G(3,6)

How to compute the Schubert class $\sigma$$^2$$_2$$_1$ in the Grassmannian G(3,6)? I remember the result is $\sigma$$_3$$_3$ + 2$\sigma$$_3$$_2$$_1$ + $\sigma$$_2$$_2$$_2$.
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Grassmannian manifold and corresponding vector field

I am assuming the definition of Grassmannian is known. Reference is Vector bundles and K-theory page no 28. I am trying to prove that the map $p:E_n(\mathbb{R}^k)\rightarrow G(\mathbb{R}^k)$ is a ...
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Plücker embedding - Two definitions

While reading several papers on the topic of the Grassmannian, I cam about two definitions of the Plücker embedding. One given as $$ \varphi: \mathbb{A}^{n \cdot d} \rightarrow \mathbb{P}^{\binom{n}{d}...
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Affine cone over the Grassmannian

I'm currently working on understanding the affine cone over the Grassmannian, which according to my paper is given by $$\text{Spec}(K[p_{ij}^{\pm} : ij \in \binom{\lbrack n \rbrack}{2} ] / I_{2,n}).$$...
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Chern classes of tangent bundle over the Grassmannian G(2,4)

What are the Chern classes of the tangent bundle $\tau_G$ of the Grassmannian $G=G(2,4)$ of lines in $\mathbb{P}^3$? This is Exercise 5.37 on page 191 of 3264 & All That by Eisenbud and Harris. ...
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Is there a moduli space $\mathcal{M}$ of the closed, $n$-dimensional, simply connected manifolds?

Is there a moduli space $\mathcal{M}$ of the closed, $n$-dimensional, simply connected manifolds with the following properties: (i) Two abstract simply connected closed manifolds $M, N\in\mathcal{...
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Does $Graff(n,\mathbb{R}^{\infty})$ generate all $n$-dimensional closed Riemannian manifolds $M$?

How does one generate all possible $n$-dimensional simply connected closed Riemannian manifolds $M$ from the affine Grassmannian $Graff(n,V)$? Would $Graff(n,\mathbb{R}^{\infty})$ suffice? (It seems ...
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BO(-) example in Weiss Calculus

I'nm reading Orthogonal Calculus by Michael Weiss, and trying to understand example 2.7, concerning the derivatives of the functor $BO$, which sends a (finite dimensional) inner product space to the ...
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On charts of Grassmanian manifolds

Suppose that $V$ is a $K-$vector space. For each positive integer $k$ the set $G_k(V )$ := {$l ⊂ V$| $l$ is a $k$-dimensional linear subspace} is called the Grassmann manifold of $k$-planes in $V$ . ...
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Good Introductory Sources for Grassmannian, Flag, and Stiefel Manifolds

I am looking to gain a deeper understanding of the Grassman, Stiefel, and Flag manifolds but finding good introductory sources so far has eluded me. I would prefer sources which have: (1) Concrete ...
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table Betti numbers for real Grassmannians

I am looking for a table of Betti numbers for real oriented and not oriented Grassmannians. Is there some references to get this ?
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Grassmann manifold (example Abraham, Marsden, Ratiu book)

I am reading, the book Manifolds, Tensor analysis of Abraham, Marsden and Ratiu. In particular, there are several points that I do not understand about a Grassmann manifold. The example starts like ...
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How a Grassmanian Parameterizes a Vector Space

Beginner question. From my understanding roughly, the Grassmannian $Gr(k, V)$ is a space which parametrizes all k-dimensional linear subspaces of the n-dimensional vector space $V$. A vector space is ...
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The Plucker relations are sufficient

Consider the Grassmannian of codimension-$d$ subspaces of a given vector space $E$ (over an arbitrary field), which I will define as $$ \operatorname{Gr}^d(E) = \{\text{linear surjections } \sigma: E \...
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352 views

Incidence correspondence of Grassmannian is a projective variety

I'm working the following question: Let $$\Sigma = \{(L, p) \in G(k,n) \times \mathbb{P}^{n-1} \mid L\subset \mathbb{P}^{n-1}, p \in L\}.$$ Here we're viewing $G(k,n)$ as $(k-1)$-dimensional ...