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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1answer
318 views

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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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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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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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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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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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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What are the first 8 homotopy groups of the complex Grassmannian $G_\mathbb{C}(2,4)$?

I was trying to find any information about the first couple of homotopy groups of the complex Grassmannian $G_\mathbb{C} (2,4)$ of complex planes in complex $4$-space. I need the first seven or eight ...
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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~ \exp\Bigg[\frac{1}{2}(\theta,...
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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 $k$-...
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Second homotopy group of real Grassmannians $\textrm{Gr}(n,m)$, special case $n=m=2$ not clear.

I have been considering real Grassmanians $$\textrm{Gr}(n,m)=O(n+m)/O(n)\times O(m)$$ appearing in certain condensed matter physics context (space of real flat-band Hamiltonians $Q(k)$ with $n$ ...
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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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Difference between Grassmann and Stiefel manifolds

I'm working on an optimization problem on manifolds and I'm having a bit of a conceptual issue with choosing between the Grassmann and Stiefel manifolds. Grassmann(2, 3) is the linear subspace of ...
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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 let ...
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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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1answer
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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 W\...
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Is the determinant bundle the pullback of the $\mathcal O(1)$ on $\mathbb P^n$ under the Plücker embedding?

Let $V$ be a $n$-dimensional complex vector space and consider the Grassmannian of complex $k$-planes $Gr(k,V)$. The Plücker embedding is an embedding $p:Gr(k,V) \to \mathbb P^M$ where $M = \left(\...
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Four-point condition for tree distances: is there a detropicalization proof?

Theorem 1. Let $G$ be a tree. Let $x$, $y$, $z$ and $w$ be four vertices of $G$. For any two vertices $s$ and $t$ of $G$, let $d \left(s, t\right)$ denote the minimum length of a path from $s$ to $...
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1answer
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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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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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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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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? $\{(l,...
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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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Derivative of Gauss map is the second fundamental form

I have been messing around with Grassmannians lately. Let $M^k\subseteq \Bbb R^n$ be an embedded submanifold equipped with the induced Riemannian metric, and consider the Gauss map $G\colon M \to {\rm ...
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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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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 the ...
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Classifying Space for What is the Infinite Unitary Group?

There is the well known result that $$ \left[X\to Gr_n\left(\mathbb{C}^{\infty}\right)\right] = Vect_n(X)$$ That is, homotopy classes of maps from a topological space $X$ into the $n$-Grassmannian are ...
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1answer
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The associated Schubert variety of a flag of subspaces of a vector space.

Let $V$ be a vector space and $W_1 \subsetneq W_2 \subsetneq ... \subsetneq W_\ell \subsetneq V $ a flag of subspaces. The associated Schubert variety is defined as : $ \Omega ( W_{ \bullet } ) = \{ \...
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1answer
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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 $\bigwedge^k V$....