# 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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### 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 ...
$\DeclareMathOperator{\Span}{span} \newcommand{\R}{\mathbf R} \newcommand{\mc}{\mathcal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\grassman}{GR} \newcommand{\set}{\{#1\}} \... 1answer 394 views ### What are the first 8 homotopy groups of the complex Grassmannian$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 ... 2answers 171 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 $$I(A) =\int d\theta_1d\theta_2~ \exp\Bigg[\frac{1}{2}(\theta,... 4answers 2k views ### Looking For a Neat Proof of the Fact that the Grassmannian Manifold is Hausdorff \newcommand{\R}{\mathbf R} Let V be an n-dimensional vector space and k be an integer less than n. A k-frame in V is an injective linear map T:\R^k\to V. Let the set of all the k-... 2answers 717 views ### Second homotopy group of real Grassmannians \textrm{Gr}(n,m), special case n=m=2 not clear. I have been considering real Grassmanians$$\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 ... 1answer 1k views ### 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 ... 1answer 1k views ### 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 ... 1answer 1k views ### 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 ... 2answers 89 views ### 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 ... 1answer 273 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 W\... 1answer 497 views ### 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(\... 1answer 723 views ### 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 ... 1answer 61 views ### 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} ... 1answer 278 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. 1answer 132 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 ... 1answer 180 views ### 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,... 1answer 155 views ### 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 ... 0answers 104 views ### 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 ... 0answers 70 views ### 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 ... 0answers 253 views ### 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 ... 1answer 209 views ### 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 ... 1answer 153 views ### 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 } ) = \{ \...
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 $\bigwedge^k V$....