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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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Proof of the Euler characteristic of the real Grassmannian $\mathbf{G}(k, n)$

I'm interested in proving the following statement, Let $\textbf{G}$(k,n) the real grassmannian and $\chi_{n,k} := \chi(\textbf{G}(k,n))$, where $\chi$ is the Euler characteristic, then $$\chi_{k,n} = \...
Fernando Avilés's user avatar
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The Grassmannian $G(k,n) \subset G(k,n+1)$ as the zero locus of the tautological sub-bundle

Following my question Pushforward of structure sheaf with respect to canonical inclusion of Grassmannian I have a follow up question I would like to ask. I would have put this as a comment, but I feel ...
Sunny Sood's user avatar
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Pushforward of structure sheaf with respect to canonical inclusion of Grassmannian

Let $j: Gr^{k,n} \rightarrow Gr^{k,n+1}$ be the canonical inclusion of Grassmannians (working over a field for simplicity). I am interested the pushforward $j_{*}\mathcal{O}_{Gr^{k,n}}$. Specifically, ...
Sunny Sood's user avatar
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Minimum subset of the Grassmanian that covers all of a vector space $\mathbb{F}_d^n$

Consider a finite field $\mathbb{F}_d$ of order $d$, and let the vector space $V=\mathbb{F}_d^n$. Let $\mathbf{Gr}(m,V)$ be the Grassmanian containing all subspaces in $V$ of dimension $m$. Suppose $S\...
Damalone's user avatar
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Dimension of the Hilbert Scheme of conics

I am trying to compute the dimension of the Hilbert Scheme of conics in $\mathbb{P}^4$ $Hilb_{2T+1}(\mathbb{P}^4)$. I started with conics lying on a plane, so, taking the ideal $I=(Q,H_1,H_2)$ for two ...
Gowexx's user avatar
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Geometric fixed-points of MO

Recall that the value of the orthogonal spectrum $\mathbf{MO}$ at an inner product space $V$ is the Thom space of the tautological bundle over the Grassmannian of $|V|$-demensiomal planes in $V\oplus ...
yifan's user avatar
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Details on: Flag incidence variety is a projective variety

Let $0 \leq p \leq q \leq n$. Define the flag incidence variety $$\text{Fl}(p,q,n):=\{(V,W) : V \leq W, V \in \text{G}(p,n), W \in \text{G}(q,n)\}$$ where $\mathrm{G}(i,n)$ is the Grassmannian, i.e. ...
Flynn Fehre's user avatar
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Generic planar sections of a projective variety

In an article by Broberg and Salberger, it is stated that The set of pairs $(\Lambda,F)\in\mathbb G(k,n)\times \mathbb P\left(\mathbb Q_d[X_0,\ldots,X_n]\right)$ for which $\Lambda\cap V(F)$ is ...
Simon Pitte's user avatar
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About the projection of a variety from a generic plane being birational

Take $Z\subseteq \mathbb P^N$ an irreducible subvariety of dimension $m$. For $\Lambda$ a projective subspace of dimension $N-m-2$ not intersecting $Z$, define $$\rho_\Lambda:Z\rightarrow\mathbb G(N-m-...
Simon Pitte's user avatar
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Why the tautological bundle of the Grassmannian has only zero as global sections?

I'm in a course on Complex Geometry, and I have been studying the Grassmannian $G_r(\mathbb{C}^N)$ as a complex manifold and the construction of its tautological bundle. As in the case of the ...
ayphyros's user avatar
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The universal bundle $\gamma_k\rightarrow BO_k$ is a real vector bundle

For all $n$, let $\gamma_k^n$ bet the tautological bundle over $Gr_k(\mathbb R^n)$, i.e. $$\gamma_k^n=\{(V,v):V\in Gr_k(\mathbb R^n), v\in V\}$$ This is also naturally identified with the associated ...
Chris's user avatar
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What is the action of $O(k)$ on $V_k(\mathbb R^n)$ making it a principal bundle?

Let $V_k(\mathbb R^n)$ be the Stiefel manifold of ordered $k$-tuples of vectors in $\mathbb R^n$. I have seen in many places that $V_k(\mathbb R^n)$ is an $O(k)$ principal bundle over the Grassmanian ...
Chris's user avatar
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Blowup of diagonal in $\mathbb{P}^r \times \mathbb{P}^r$

Let $X= \mathbb{P}^r \times \mathbb{P}^r$. Suppose we blowup $X$ along $\Delta$ the diagonal to get $\tilde X$. I want to show that this is isomorphic to the fibre product which I describe below - Let ...
Angry_Math_Person's user avatar
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Can we construct the exterior algebra just from simple multivectors?

$ \newcommand\K{\mathbb K} \newcommand\Ext{\mathop{\textstyle\bigwedge}} \newcommand\Lip{\mathrm{Lip}} \newcommand\ev{\mathrm{ev}} \newcommand\Gr{\mathrm{Gr}} $Let $V$ be a finite-dimensional $\K$-...
Nicholas Todoroff's user avatar
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Induced morphism of smooth variety to the Grassmannian on global sections?

In Huybrechts & Lehn, The Geometry of Moduli Spaces of Sheaves, page 143, it reads Let $X$ be a smooth variety. Suppose $E$ is a locally free sheaf of rank $r$ which is generated by its space of ...
Anthony Lee's user avatar
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$H^i\left(BO_m ; \mathbb{Z}_2\right) \cong H^i\left(B O_n ; \mathbb{Z}_2\right)$ for $i \leq m$ and $i \leq n.$

The notes I am reading say that groups in the title are isomorphic. Could someone explain to me why it is the case? Here by $BO_n$ I mean the infinite Grassmann manifold of $n$-dimensional subspaces. ...
Haldot's user avatar
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Dimers from Postnikov diagrams

I am reading several papers on Postnikov diagrams, dimer models, and quivers. From the Postnikov diagram, we can draw a dimer model and an ice quiver. My question is as follows. Does every dimer model(...
aleph0's user avatar
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Show that Mat$_{n\times k}^k\to G_k(\mathbb{R}^n)$ is a fibration

I am trying to show that Mat$_{n\times k}^k\to G_k(\mathbb{R}^n)$ is a fibration where Mat$_{n\times k}^k$ denotes the full-rank matrices of rank $k$. I know that $V_k(\mathbb{R}^n)\to G_k(\mathbb{R}^...
Michael Wang-Wakamatsu's user avatar
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1 answer
148 views

Hausdorff property of Grassmannian

Good evening to everyone. I am new to Manifold Theory, so I am trying the last weeks to study some chapters from the book of John M. Lee 's Introduction to Smooth Manifolds. I was trying to understand ...
Petros Karajan's user avatar
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Connections between an identity about minors and Plücker relations

Consider a $2 \times 4$ matrix $A = \begin{bmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ \end{bmatrix}$. Its all minors of order 2, such as $A_{13} = \begin{...
YSouSerious's user avatar
3 votes
2 answers
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Calculate the Stabilizer of an element in the Grassmannian

I know there is a virtually same question and its answer on this site, but neither the questioner nor the answerer appears to be active at the moment. Define a group action on Grassmannian $$\begin{...
一団和気's user avatar
2 votes
1 answer
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Manifold of subrepresentations of an isotypic subspace

Let $ \Theta: G\to GL(V) $ be an irreducible complex representation of a Lie group $ G $. Then consider $ \bigoplus_\mu \Theta: G \to GL(\bigoplus_\mu V) $, the direct sum of $ \mu $ copies of $ \...
Ian Gershon Teixeira's user avatar
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Prove $\Lambda = \{ (\text{Pl}(L), p) \in \text{Pl}(\text{Gr}(k,n))) \times Z \ | \ p \in L\}$

My problem Let $Z$ be a projective variety. I have to prove that $\Lambda = \{ (\text{Pl}(L), p) \in \text{Pl}(\text{Gr}(k,n))) \times Z \ | \ p \in L\}$ is a projective variety. Here $\text{Gr}(k,n)$...
Oopsilon's user avatar
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Show that the submersions $V_k(\mathbb R^n) \to G_k (\mathbb R^n)$ and $O(n) \to G_k (\mathbb R^n)$ are fibrations

Show that the submersions $F: V_k(\mathbb R^n) \to G_k (\mathbb R^n)$ and $G: O(n) \to G_k (\mathbb R^n)$ are fibrations. Deduce that $\text{Mat}^k_{n \times k}(\mathbb R) \to G_k(\mathbb R^n)$ is a ...
Squirrel-Power's user avatar
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1 answer
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Show that the Grassmanian is a manifold using Godement's theorem on $O(n)$

I am trying to prove that the Grassmannian $G_k(\mathbb R^n)$ is a manifold using Godement's theorem on $O(n)$. I know that $G_k(\mathbb R^n) = O(n) / (O(k) \times O(n-k))$, but I am not sure how I ...
Squirrel-Power's user avatar
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Pullback of the tautological subbundle under the canonical inclusion

Similar questions seem to have been asked on the stack exchange before, so I apologise for any repetition, but none of the responses seem to answer the question in satisfactory detail, at least for me....
Sunny Sood's user avatar
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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 \...
吴yuer's user avatar
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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 $\tau^*\colon \mathsf{H}^*(G_{m+...
Frank's user avatar
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Exercise 5.4.1 in Smith's Invitation to Algebraic Geometry; lines that are tangent to conic are closed subvariety of Gr$(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 ...
Neckverse Herdman's user avatar
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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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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 ...
Laurence PW's user avatar
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2 answers
126 views

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 ...
red_trumpet's user avatar
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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 ...
Frank's user avatar
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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 ...
Marcos Martínez Wagner's user avatar
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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 ...
Richard's user avatar
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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 ...
Skadiologist's user avatar
1 vote
1 answer
141 views

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 ...
Alex's user avatar
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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-...
Leonardo's user avatar
1 vote
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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 ...
MMH's user avatar
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3 votes
1 answer
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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 ...
Dario Antolini's user avatar
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59 views

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 ...
fr_andres's user avatar
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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 ...
Bence Racskó's user avatar
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356 views

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, ...
yi li's user avatar
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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\...
Cellardoor's user avatar
1 vote
1 answer
73 views

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:...
s.h's user avatar
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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 ...
Paweł Czyż's user avatar
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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 $...
Conjecture's user avatar
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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 $\...
user267839's user avatar
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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 $...
BulkyMolaMola's user avatar
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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 ...
Matt Park's user avatar

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