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 $r$.

Filter by
Sorted by
Tagged with
1
vote
1answer
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 ...
2
votes
0answers
67 views

Prove a “distance” is a metric between vector spaces

In a paper by Alan Edelman, "The geometry of algorithms with orthogonality constraints" (page 35 ), there are several definitions to the notation "distance" between vector spaces on the Grassmanian. ...
1
vote
1answer
311 views

Tangent bundle to Grassmannian

Let $G :=G(k,n)$ be the Grassmannian of $k$-planes in an $n$-dimensional vector space. We automatically have the exact sequence for the universal (tautological) bundle $\mathcal{S}$: $$0 \to \mathcal{...
2
votes
1answer
171 views

Grassmannian $G(2,3)$ homeomorphic to the projective plane $\mathbb{P}_{\mathbb{R}}^2$

I'm studying the Grassmannian for low dimensions and I saw that $G(2,3)\cong \mathbb{P}_{\mathbb{R}}^2$. Reference: https://en.wikipedia.org/wiki/Grassmannian Mostly, I understand the intuitive idea,...
0
votes
1answer
205 views

Definition of k-plane as linear subspace of dim. k?

I am currently reading L.W. Tu's "Introduction to Manifolds" (2nd ed.). In exercise 7.8, which is concerned with showing that the Grassmannian $ G(k,\mathbb R^n) $ is a smooth manifold, Tu refers to ...
0
votes
0answers
94 views

Schubert variety associated to a flag of subspaces of a vector space

At the end of the following page : The associated Schubert variety of a flag of subspaces of a vector space. , the author says : Let $[W]\in X=X_{\underline i}$, by construction: \begin{equation} [W]=...
1
vote
0answers
32 views

Iwasawa decomposition for real Grassmannians

I am interested in the Lie algebra decompositions of the isometry group of the real Grassmanianns (seen as symmetric spaces). More precisely, let $Gr(r,n)=O(n)/(O(r)\times O(n-r))$. Can someone ...
2
votes
0answers
56 views

Generically rebuilding non-degenerate matrices from a vector

Let $X$ be the subspace of $3 \times 2$ complex matrices with full rank, where we identify matrices differing from one another by a scalar. There are some natural maps from $X$ into $\mathbb{C}\!\...
2
votes
0answers
68 views

How can I find the subspace basis most “aligned” to the ambient space axes?

Let's say I have a 4-dimensional subspace of a 300 dimensional ambient space, represented by an orthonormal basis matrix $A$. What I essentially want is to rotate this basis matrix, so that the ...
2
votes
0answers
133 views

Bundles on Grassmannians

Let $V,W$ be two $4$-dimensional vector space over a field, $G(2,V),G(2,W)$ the Grassmanninas of planes in $V$ and $W$, $S_V$, $Q_V$ the tautological subbundle and quotient bundle of $G(2,V)$, and $...
3
votes
0answers
82 views

A smooth map from $Gr_{\mathbb{C}} (1,2)$ to $Gr_{\mathbb{R}} (2,4)$

$\mathbf{Problem}$: Let $Gr_{\mathbb{C}} (1,2)$ be the complex Grassmannian manifold that consists of all the complex lines going through the origin in $\mathbb{C}^2$, and $Gr_{\mathbb{R}} (2,4)$ be ...
1
vote
0answers
109 views

Minimization over constrained projection matrices

Is there an elegant solution to the following minimization problem? $$ \min_{B} \left( \sum_i a_i^\top C_i a_i \right) - \left( \sum_i C_i a_i \right)^\top \left( \sum_i C_i \right)^{-1} \left( \...
2
votes
1answer
137 views

Reference for Grassmannian Manifold

I need to study Grassmannian manifold in a good way, like study vector bundles, tangent bundle,... and etc on Grassmannian manifold. I found some lecture and books but they were written an ...
1
vote
1answer
109 views

Best way to show that this is a subvariety of the Grassmannian

Let $k$ be a field and let $\text{Gr}(r,k^n)$ be the Grassmannian of $r$-dimensional subspaces of $k^n$. Fix a linear map $\phi\colon k^n \rightarrow k^n$. Define $X_{\phi}$ to be the subspace of $\...
3
votes
0answers
142 views

Proof the Grassmannian is a local functor?

The Grassmannian functor $\mathrm{Gr}_{n, r}$ sends a ring $A$ to the set of rank $n$ summands of the free module $A^{n + r}$. This is a local functor and I.1.3.13 of Demazure and Gabriels book "...
3
votes
0answers
88 views

Ricci tensor of the Grassmannian manifold

I'm wondering if anyone could help me with calculating the Ricci tensor for the Grassmannian manifold. For Kähler manifolds we have: \begin{equation} R_{\mu \bar{\nu}} = -\partial_{\bar{\nu}} \...
1
vote
0answers
33 views

Is the set of subspaces generated by positive vectors a closed subset of the Grassmanian?

Consider the spaces $G=\{V = (v_1|\dots|v_k)\,:\,v_i \in\mathbb{R}^n,\,\mathrm{rank}(V) = k\}$ and $G_+=\{V = (v_1|\dots|v_k)\in G\,:\,v_i \in[0,\infty)^n\}$. If $Gr(k,n)$ denotes the real Grassmanian ...
1
vote
1answer
539 views

matrix wise tangent inverse (arctan)

Given a matrix $X$, an expression for the matrix cosine and sine are given by $$ \textrm{cos}(X) = \frac{e^{iX} + e^{-iX}}{2}\\ \textrm{sin}(X) = \frac{e^{iX} - e^{-iX}}{2i} $$ I have been trying to ...
5
votes
1answer
227 views

What is the amplituhedron?

I would like to know what exactly this object is without any nasty physics stuff. The wikipedia says that it "generalizes the idea of a simplex in projective space." It's easy for me to see how the ...
1
vote
0answers
61 views

Grassmann space in Minkowski space

Let $V$ be a real vector bundle with non-degenerate metric whose index is $(p,q)$, i.e. $p$ ($q$) is the dimension of maximal subspace of positive(negative) definite. Let $G^+_p(V)$ be the all ...
0
votes
1answer
35 views

Relationship between Cross-cap and Grassman Manifold

My professor, said that https://en.wikipedia.org/wiki/Cross-cap#/media/File:CrossCapTwoViews.PNG is a visualization of Grassmann Manifold of n=3,d=1. Can anyone help me understand this please.
0
votes
0answers
31 views

Grassmanian is a coherent space

How to prove that Grassmanian is a coherent space? I was thinking about it and I can't understand how to build a path from one subspace to another, if each subspace is determined by the matrix and ...
9
votes
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 ...
3
votes
1answer
368 views

Structure of complex Grassmannian $\textrm{Gr}_\mathbb{C}(2,2)$

I recently asked a question about the topology of real Grassmannian $$\textrm{Gr}_\mathbb{R}(2,2) = \frac{O(4)}{O(2)\times O(2)},$$ see Second homotopy group of real Grassmannians $\textrm{Gr}(n,m)$, ...
0
votes
1answer
253 views

What is the Eilenberg-MacLane space $K(\mathbb Z_2, 2)$?

Question 1: I know that $K(\mathbb Z, 2)$ is $\mathbb CP^\infty$ and that $K(\mathbb Z_2, 1)$ is $\mathbb RP^\infty$. But, how about $K(\mathbb Z_2, 2)$? Do we similarly have a good description of ...
4
votes
2answers
312 views

How to define a topological space of straight lines?

Today in a differential geometry lecture, the lecturer put down a question for us to think about: Given a regular curve (differentiable function $\alpha:I\to\Bbb R^3$ on an open interval with $\alpha'...
1
vote
0answers
78 views

How can I get the matrix form for a schubert cell?

I am trying to learn how to understand the cell-decomposition for the grassmannian and am following these notes: http://www.math.drexel.edu/~jblasiak/grassmannian.pdf. On page 2 the author considers ...
0
votes
1answer
117 views

Is there a description of the Grassmannian as a homogeneous space where the principal bundle is one associated to the universal vector bundle?

Consider the Grassmannian $\text{Gr}_{n,k}(\mathbb{F})$ for $\mathbb{F}\in \{\mathbb{R},\mathbb{C},\mathbb{H} \}$. Does there exist a presentation of it as a homogeneous space $$ B \hookrightarrow G \...
1
vote
0answers
128 views

Does the embedding of the Grassmannian in the projectivization of the exterior algebra require the axiom of choice?

According to the wikipedia page for the Grassmannian, the embedding in the projectivization of the exterior algebra of $V$ requires the choice of, for every r-dimensional subspace, a basis for that ...
1
vote
1answer
110 views

Identification of Grassmannian manifolds

This is a problem on Milnor's Characteristic Classes. It asks me to show that $G_n(\mathbb{R}^m)$(all $n$-planes in $\mathbb{R}^m$) is diffeomorphic to the manifold consisting of all $m\times m$ ...
1
vote
0answers
74 views

Grassmannians as functors on *all* vector spaces and injective maps

Given a field $K$ and a vector space $V$ of dimension at least $1$ we can define the projective space $PV=(V - \{0\})/K^\times$ to be the quotient of $V$ by the action of the multiplicative group of $...
0
votes
2answers
126 views

Exterior product and four vectors?

Quite a straight forward question but something I don't really understand. I have a mediocre grasp on the concept of exterior products in $3$ dimensions. However, is the product defined for two $4$-...
1
vote
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 ...
1
vote
1answer
68 views

Geometric interpretation of lines in a plane.

In projective geometry, there is a nice way to see that the set of lines through a point $p$ in $\Bbb{P}^3$ is naturally a projective plane: Simply choose a plane $H$ that does not contain $p$. Then ...
1
vote
0answers
68 views

Is every $n$-dimensional linear subspace of $\mathbb{R}^{2n}$ a Lagrangian submanifold?

Let $\mathbb{R}^{2n}$ have the standard symplectic form $\omega = dx_1\wedge dy_1 + \cdots + dx_n\wedge dx_n$. If we consider an $n$-dimensional subvector space of $\mathbb{R}^{2n}$ as a submanifold, ...
1
vote
0answers
255 views

Geometry of tangent space to Grassmannian

In considering what the dimension of a Grassmanian, $G(k,V)$, is, I came across an interesting property of the tangent space to a Grassmanian. Specifically, given $\mathbb{P}E \subset G(k,V)$, where $...
3
votes
0answers
246 views

Plücker coordinates and grassmannians

Let $X$ denote the set of lines in $\mathbb{P}_k^3$ and $\varphi:X\rightarrow\mathbb{P}_k^5$ be the morphism that sends, once we have chosen a reference, each $(a_0:\cdots:a_3)\vee(b_0:\cdots:b_3)$ to ...
2
votes
2answers
322 views

Why graph of a linear map, gives a bijection in the proof of Grassmannian as a manifold?

In the proof of Grassmannian $(G_k(\mathbb{R}^n))$ as a manifold, we take an open neighbourhood $U_L = \{L^\prime\;|\; L^\prime\cap L^\perp=\{0\}\}\subset G_k(\mathbb{R}^n)$ of a $k$-dimensional ...
3
votes
1answer
715 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 $...
2
votes
0answers
189 views

Prove that the plucker mapping is an immersion

I am considering the Plücker map \begin{align*} \Phi:\mathbb{G}_k(\mathbb{C}^N)&\longrightarrow\mathbb{P}(\bigwedge^{\quad k}\mathbb{C}^N)\\ \langle z\rangle &\longmapsto\langle z_1\wedge\...
8
votes
2answers
710 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 $n$ ...
2
votes
1answer
279 views

Intersection of Schubert cycles

I want to compute intersection of the Schubert cell $\sigma_{(3,0)}$ with all the cells $\sigma_{a_1, a_2}$ in the grassmanian $G(2,5)$. I am not sure I am doing correctly but I can't see my mistake. ...
2
votes
1answer
531 views

Understanding the cohomology ring of the Grassmannian

Some background first: I'm trying to understand the solution of some enumerative geometry problems, such as proving that a smooth cubic contains $27$ lines. I know that this becomes easier once one ...
2
votes
0answers
51 views

Topology on the (fibre?) bundle of decomposable $m$-tangent vectors

Let $V$ be a vector space, then denote the simple or decomposable $m$-vectors in $\bigwedge^m V$ by $GC_m(V)$. I am struggling to understand the topology of the bundle $GC_m(TN) = \bigsqcup_{q\in N} ...
1
vote
1answer
55 views

Ref. Request — Tautological Bundle over $G_{k,n}(\mathbb{R})$

I'm interested right now to learn more about the tautological bundle over the Grassmann manifold $G_{k,n}(\mathbb{R})$, but I'm currently having trouble finding the appropriate texts to check out. ...
2
votes
0answers
76 views

Chern classes of hyper-Kähler fourfolds in Grassmannians

Both the Fano variety $F$ of lines in a general cubic fourfold and the "Debarre–Voisin" fourfolds $Y_\sigma$ introduce in [DV] are smooth, four-dimensional, hyper-Kähler subvarieties of Grassmannians (...
2
votes
1answer
389 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 ...
3
votes
2answers
655 views

Sub Matrix of an Orthogonal Matrix is always singular?

I am trying to implement Grassmanian rank one update (GROUSE) as per this paper . For the solution of the least squares problem of (1) in the paper the value of $w$ is $w$ = $(U_{\Omega_{t}}^TU_{\...
2
votes
0answers
225 views

Why is the second homotopy group of the complex Grassmannian not trivial?

I am coming from a physics background, and reading through different articles on topological classification. In one article (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.51), they ...
1
vote
1answer
659 views

Construction of Grassmannian

I am trying to understand what is a Grassmannian. Starting with the projective space $\mathbb{R}P^n$ = {lines in $\mathbb{R}^{n+1}$} and the grassmannian $G_r(k,n)$ = {$k-dimentional\space subspace\...