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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What is the meaning of this statement in the proof that grassmanian is a manifold?
In wiki page https://en.wikipedia.org/wiki/Grassmannian of Grassmannian, in the endowment of smooth structure to Grassmannian, I encounter this statement,
"For each ordered set of integers $1 \...
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Cohomology of product Grassmann manifolds
For infinite complex Grassmann manifolds, we always have embedding $\tau\colon G_m\times G_n\to G_{m+n}$, then how to prove the induced homomorphism of cohomology rings $\tau^*\colon \mathsf{H}^*(G_{m+...
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Exercise 5.4.1 in Smith's Invitation to Algebraic Geometry; lines that are tangent to conic are closed subvariety of Gr$(2,3)$ in $\mathbb{P}^2$
I am trying to solve Exercise 5.4.1 in Karen E. Smith's Invatation to Algebraic Geometry:
Fix an irreducible conic $C$ in $\mathbb{P}^2$. Show that the set of lines in $\mathbb{P}^2$ that fail to ...
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If all the vector spaces in a convergent sequence in the Grassmanian contains a vector, then the limit of the sequence contains the vector?
Let $G(k,n)$ denote the Grassmanian of $k$-dimensional vector subspaces in $\mathbb R^n$. Suppose $W_n$, $n \in \mathbb N$ is a sequence of subspaces in $G(k,n)$ converging to some $W$ in its topology....
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How to determine Plucker coordinates of Infinite dimension Sato Grassmannian? [closed]
I wish to determine Plucker coordinates of Infinite dimension Sato Grassmannian. I've been reading the original paper by Sato and other related works, but couldn't understand few things
Do Plucker ...
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Map from Orthonormal Frame to Grassmannian is Quotient Map
Let $\text{Or}_k(\mathbb{R}^n)$ be set of orthonormal $k$-frames for $k\leq n$ (collection of set of orthonormal $k$ vectors in $\mathbb{R}^n$) and let $\text{Gr}_k(\mathbb{R}^n)$ be Grassmannian of ...
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How to see the second fundamental form is symmetric when defining it as the derivative of the Gauss map?
Let $X \subset \mathbb C^N$ be an $n$-dimensional complex submanifold.
Voisin writes in [1]
Let us introduce the second fundamental form
$$\Phi: S^2 T_{X,x} \to \mathbb C^N / T_{X,x}$$
which can be ...
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Proving the Grassmannian is a manifold
The Grassmannian $\mathrm{Gr}(k, V)$ is the space of all $k$-dimensional subspaces of a (real or complex) vector space $V$. I am interested in the following strategy of proving $\mathrm{Gr}(k, V)$ is ...
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Is the Grassmannian a Fiber Bundle?
Let $V$ be a real, finite dimensional, vector space, let $\mathrm {Ind}_k(V)$ be the set of all linearly independent vectors $(v_1,...,v_k)\in V^k$ and let $\pi :\mathrm {Ind}_k(V) \rightarrow \mathrm ...
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Is $\operatorname {GL}(n)\times \operatorname{Gr}(m,n)\to \operatorname{Gr}(m,n)$ closed?
Suppose $k$ is a field, and $m<n$ are nonnegative integers. Let $\operatorname{Gr}(m,n)$ be the Grassmannian (whose points are $m$-dimensional subspaces of a $n$-dimensional linear space). Then we ...
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The universal sub-bundle on Gr(2,4) is stable and exceptional
The universal sub-bundle and the dual of the universal quotient bundle on $\operatorname{Gr}(2,4)$ are specific examples of spinor bundles on a quadric. They arise from an irreducible representation ...
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How do I view the tangent bundle of 2-sphere as pullback from the universal bundle?
I am trying to visualize that any vector bundle is a pullback from the Tautological bundle of Grassmannian.
If the tangent bundle of $S^2$ is not a simple example, is there a/the simplest example? The ...
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Example 1.36 in Lee's Introduction to smooth manifold.
Let $V$ be an n-dimensional real vector space decomposed as a direct sum: $V=P\oplus Q$, with $\dim P=k$ and $\dim Q=n-k$. The graph of any linear map $X:P\rightarrow Q$ can be identified with a k-...
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Generating Random Linear Subspace
Let $n$ and $m$ be two integers such that $m\leq n$. Let $G_{n,m}$ be the set of all m-dimensional linear subspaces of $\mathbb{R}^n$. Assume we want to generate a subspace of dimension $m$ which is ...
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Plücker relations in Sagemath from Macaulay2
I am trying to implement the Plücker relations in Sagemath. Sage has an interface for Macaulay2, and this latter has a command Grassmannian(k-1, n-1) for computing ...
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Expected value of the projective metric between random orthogonal Stiefel matrices in $\mathbb{R}^{N \times k}$ equals $1 - \frac{k}{N}$
This is a call for help to the random-matrix-theory savvy people. I've observed the below equality in experiments, and have been looking for a proof in the RMT literature but couldn't find one. I'd ...
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de Rham cohomology of the Grassmannian bundle
Let $X$ be a smooth $n+m$ dimensional manifold and $\pi:Y\rightarrow X$ the bundle whose fibre over any $x\in X$ is the Grassmann manifold of all $m$ dimensional subspaces of $T_xX$. This is ...
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What does the universal property of Grassmannian mean?
I was trying to figure out what is the universal property of Grassmannians, I saw a similar question Universal property of the Grassmanian [closed].
And I have checked the references in the answer, ...
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Irreducibility of variety of intersecting linear spaces
For the purposes of my research, I want to know if for any choices of dimensions $d_i,d_I\in \mathbb N\cup\{-1\},$
$$\mathcal V:=\{(L_1,\ldots,L_m):L_i\in \mathrm{Gr}(d_i,\mathbb P^n), \dim L_I\ge d_I\...
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What is the geometry of $L=\lbrace W \subset V:\text{$W$ is Lagrange subspace of $V$} \rbrace$?
I know the following statement:
Theorem. Let $(V,\omega)$ be a finite dimensional symplectic linear space. Then the symplectic group $Sp(V)$ of $V$ transitively acts on the set $L=\lbrace W \subset V:...
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The space of direct decompositions
The Grassmannian $\mathrm{Gr}(k, n) = O(n) / O(k)\times O(n-k)$ describes all $k$-dimensional subspaces of $\mathbb R^n$. The product space $S=\mathrm{Gr}(k, n)\times \mathrm{Gr}(n-k, n)$ represents ...
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When determinant bundle is very ample
For a vector bundle $V$ on a projective variety $X$, let $\Bbb P(V) $ be the projective bundle of hyperplanes. Call $V$ a very ample if $\mathcal O_{\Bbb P(V)}(1)$ is very ample on $\Bbb P(V)$.
Let $...
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Graph of Rational map
Let $X, Y$ be projective irreducible varieties over $k=\mathbb{C}$ and
$\varphi: X \dashrightarrow Y$ a rational map. Let $U \subset X$ the maximal open dense subset $U \subset X$ of points where $\...
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Why is this not a Postnikov diagram?
I am following this paper on Grassmannians and Cluster Structures. I drew the following diagram for $Gr(2,6)$:
However, it doesn't satisfy the property that each alternating region is labelled by a $...
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An invariant that determines membership in an element of $Gr(2,\mathbb{C}^3)$
Consider two basis vectors for a two-dimensional subspace of $\mathbb{C}^3,$ $v_1=(z_1,z_2,z_3),v_2=(z'_1,z'_2,z'_3)$. I am looking for a geometric or algebraic invariant that determines if another ...
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Diffeomorphism between $SO(3)$ and $\mathbb{R}P^3$
I am curious if one can find a diffeomorphism between $SO(3)=\{A\in M_3(\mathbb{R}): AA^T=I_3 \text{ and det(A)=1} \}$ and $\mathbb{R}P^3$ using Grassmannian manifolds. I have seen multiple proofs ...
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Intersection of schubert varieties
Let $L_1$ and $L_2$ $\in$ $\mathbb{P}^4$ be two planes that intersect in exactly one point $Q$. Let $P_1 \in L_1$, $P_2 \in L_2$ points, such that $P_1 \neq Q \neq P_2$. Using the duality theorem, ...
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Proving that the Grassmannian is a smooth manifold [duplicate]
I am just trying to come up with a simple proof showing that the Grassmannian ($Gr(k,\mathbb{R}^n)$) is a manifold.
I was referring to Lee's Introduction to Smooth Manifolds but I can't seem to follow ...
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Exterior algebra: property of signed area
The wiki on exterior algebras lists of number of properties enjoyed by the signed area, and all of them make sense except for this one:
$A(v + rw, w) = A(v, w)$ for any real number $r$, since adding ...
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Derivative of the optimal soluction w.r.t to the parameters in a Grassmann manifold optimization problem
We have a Grassmann manifold optimization problem
$$ \min_{x} f(x,\alpha),$$
where $x$ is in Grassmann manifold, $\alpha$ is a parameter. Optimal solution $x^*$ is then can be represented as a ...
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Subvarieties of Schubert varieties over finite fields
Let $\mathbb{F}_q$ be a finite fields and suppose $\Omega_{\alpha} (\ell, m)$ denote the Schubert variety given by
$$\Omega_{\alpha} (\ell, m)= \{ [P] \in G(\ell, m) : \dim (P \cap A_i) \ge i\}$$
...
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Proving that $S\mapsto\text{Hom}(\bigwedge^2S,\mathbb R)$ defines a vector bundle over a Grassmanian.
My question has to do with the most voted answer to this post. There, it is suggested that the map
$$S\in G_n(\mathbb R^{2n})\mapsto\text{Hom}(\wedge^2S,\mathbb R)$$
determines a vector bundle over ...
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smooth points on variety of linear subspaces intersecting a given subspace
$\newcommand{\Ind}{\operatorname{Ind}} \newcommand{\Gr}{\operatorname{Gr}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\R}{\mathbb{R}} \newcommand{\GL}{\operatorname{GL}} \newcommand{\codim}{\...
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Grassmannian is embedded in projective space by Plücker map
The Grassmannian $G(n,k)$ is the space of $k$ dimensional linear subspaces of $\mathbb R^n$. The Plücker map $\pi: G(n,k) \to \mathbb R P^m, m = \binom nk -1,$ is defined by sending a linear space $L$ ...
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Quotient $G/B$ of Lie group $G$ by Borel group $B$ contained in a Grassmannian
Let $G$ be a compex Lie group with semisimple Lie algebra $\mathfrak{g}$, the connected subgroup
$B$ of $G$ with Lie algebra $\mathfrak{b} := \mathfrak{b} \oplus \bigoplus_{\alpha \in R^{+}} \mathfrak{...
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$\|P_X-P_Y\|_2=\|(1-P_Y)P_X\|_2$
Let $X,Y$ be two linear subspace of the same dimension of a finite linear space.
Denote by $\|\cdot\|_2$ the induced matrix norm. The question asks me to prove the following, as written in title.
$$\|...
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Subspaces of $\mathbb{R}^n$, related to Grassmannian
This question arose when I was trying to understand a proof related to the Grassmannian of $\mathbb{R}^n$.
Let $P=\langle e_1,\ldots,e_k\rangle$ and $Q=\langle e_{k+1},\ldots,e_n\rangle$, where $\{e_1,...
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Cantor-like set in $\mathbb{R}^2$ is purely 1-unrectifiable
I was just reading about rectifiability, and an example of purely 1-unrectifiable set can be given as follows. Consider the Cantor-like set $C_{1/2}$ obtained like this:
$$[0,1] \to [0, 1/4] \cup [3/4,...
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Why isn't the dimension of the space of curvature-like tensors equal to the dimension of the Grassmannian?
It's a well known fact that if $V$ is a vector space of dimension $n$, then the vector space consisting of all curvature-like tensors (see this text for a precise definition and a proof) has dimension ...
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What is the dimension of relative Grassmannian?
Let $E$ be a $r$ rank vector bundle on a smooth projective curve $C$ of genus $g$. Let $G$ denote the Grassmannian of $(r-k)$ rank locally free quotients of $E$.
A paper by Mukai-Sakai seems to claim ...
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Grassmannian $G(m,n)$ is locally Euclidean of dimension $m(n-m)$
A generalization of projective spaces are the Grassmann manifolds. Consider $G(m,n)$ as the set of all $m$-dimensional subspaces of $\mathbb{R}^n$. The topology on $G(m,n)$ should then arise ...
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lines in projective space with non-algebraically closed field
I apologize in advance for this possibly trivial question.
Let $k$ be a field that is not necessarily algebraically closed. We consider $\mathbb{P}^n_k$ and let $l$ be a line in it. There is a ...
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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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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 $n \times p$ ...
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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/...
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