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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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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plane as a 4D anti-vector and the distance from the origin
So, we can get a plane in 4D the following way using the wedge product - p ^ q ^ r. It encompasses information on both the normal to the plane and the distance to ...
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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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Fano Variety of a Hyperplane (Harris Algebraic Geometry A First Course)
I have a question about Example 6.19 (p. 70) in Harris' Algebraic Geometry: A First Course.
There is stated that the Fano variety
$$F_k(X) := \{\Lambda \in \mathbb{G}(k,n) \ \vert \ \Lambda \subset X \...
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Irreducible decomposition of the ideal defining the middle Grassmannian
We work over the field of complex number, using the Grothendieck projectivization, that is we have
$$
\mathbb P(\cdot):=\operatorname{Proj} \bigoplus_{m \ge 0} \operatorname{Sym}^m(\cdot).
$$
Let $X$ ...
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What is a linear subspace of projective space?
I have to give a talk about Grassmannians and I have trouble understanding the following sentence:
A $k$-dimensional vector subspace of an $n$-dimensional vector space $V$ is the same
as a $k -1$ -...
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Understanding the notation on Grassmanians
I'm studying a book about Lie Group Actions, and there is a part where they talk about Grassmanians. A k-Grassmanian on vector space $V$ is defined as follows $$Gr_k(V)=\{W: W \text{ is a subset of } ...
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Show that $\mathbb{P}(V)$ and $G_{1}(V)$ are diffeomorphic
Be $V$ a real vector space of dimension $n+1$ defined over $\mathbb{R}^{n+1}$. Build up a differential structure on $\mathbb{P}(V)$, the projective space on $V$. Show that there is a diffeomorphism ...
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Deviation of Random Projections
Let $P$ be an orthogonal projection in $\mathbb{R}^n$ onto an $m-$dimensional random subspace uniformly distributed in the Grassmannian $G_{n,m}$. Let $T$ be a bounded subset of $\mathbb{R}^n$. Let $x$...
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Grassmannian gaussian integral
$A$ a 2 by 2 matrix and let it be anti symmetric. Then $detA=A_{12}^2$.
The integral is said to hold:
$$\int d\hat\theta\int d\theta e^{\theta ^T \cdot A \cdot \hat\theta}=detA$$
Where $\hat\theta ,\...
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Map between Stiefel manifold and the Grassmannian
I'm working on problem 2-7 in Lee's introduction to Riemannian Manifolds and am having trouble on part (b):
Let $V_k(\mathbb{R}^n)$ be the Stiefel manifold and $G_k(\mathbb{R}^n)$ denote the ...
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Confused about notation for the cohomology of Grassmannians
In Milnor and Stasheff's book "Characteristic Classes", problem 7b, they ask us to show that the cohomology ring $H^* (G_n (\mathbb{R}^{n+k}, \mathbb{Z}_2)$ is generated by the Stiefel-...
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How is the dimension of a Grassmannian defined?
I'm new to topology, I'm struggling with how the dimension of a (real) Grassmannian Gr(k,n) is defined?
I know the definition of dimension of a vector space, but Gr(k,n) does not seem like a vector ...
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How is Grassmann Algebra better than just using determinants?
I am reading Roger Penrose's road to reality. On page 209 ,210 he introduces the idea of Clifford algebra and how we can get back quarternion algebra by consideration of the 'second order' quantities ...
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What is $\operatorname{Hom}(S,Q^\vee)$ for a Grassmannian?
Let $G(k,V)$ be the Grassmannian of $k$-planes in a complex vector space $V$ of dimension $n$. There is the famous universal exact sequence of vector bundles on $G(k,V)$
$$
0 \to S \to V \otimes \...
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Join of two varieties construction via rational maps.
Consider the map $j: X\times Y \to\mathbb{G}(1,n), \ ([v],[w])\mapsto [v\wedge w]$ i.e. sending two points to their line where $X$ and $Y$ are two projective irreducible varieties. I would like to ...
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Non trivial lower bound for $\sup_P \|Px\|$ where $x$ is fixed, and the supremum if over all full-rank projection matrices $P \in R^{k \times n}$
Let $1 \le k \le n$ be integers and let $G_{n,k}$ be the grassmannian of $k$-dimensional subspaces of $\mathbb R^n$. For any $U \in G_{n,k}$, let $P_U$ be the orthogonal projection onto from $\mathbb ...
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Computing and integrating the Gaussian curvature if the real Grassmannian G(1,3)
I am trying teach myself the basics of curvature by doing concrete computations.
I would like to check the Gauss-Bonnet formula for the real Grassmannian $G(1,3)$ of lines in $\mathbb{R}^3$, which is ...
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Trivial extension for the tangent bundle of Grassmannian of planes in $\mathbb C^5$
Let $X$ be the 6-dimensional Grassmannian of 2-planes in a 5-dimensional vector space $V$, namely $X=G(2,5)$.
I want to compute
$$
H^1(X,\Omega_X \otimes L)
$$
where $\Omega_X$ is the cotangent bundle ...
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Grassmannians which are complete intersections
I'm looking for examples of complete intersections which are U.F.D. I know that that the coordinate ring of $Gr(k,n)$ (in its Plucker embedding) is an U.F.D. I would like to know examples of some ...
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$\gamma_k^m = (E,p,G_k(F^m))$ is locally trivial
I'm currently studying bundles from Husemaller and I'm stuck with the proof that $\gamma_k^m = (E,p,G_k(F^m))$ at p.$25$.
The thing I don't get is why $U_H$ is open and $h_H$ is an isomorphism. It ...
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Is the push-forward of the pull-back of a regular section still regular?
I refer to this answer for definitions.
Let $X$ be a smooth projective uniruled variety covered by lines, $Pic X=\mathbb Z$, and let $L=\mathcal O(1)$ (if it's simpler, $X$ is just an isotropic ...
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Dual of universal quotient bundle globally generated
We are consistent with the notation in the book of Hartshorne.
Let $X=G(\mathbb P^k, \mathbb P^n)$ be the Grassmannian parametrizing $\mathbb P^k$ contained in $\mathbb P^n$. We have the so-called ...
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How to derive the Leibniz rule for the Grassmann number?
Consider the Grassmann number $$\{\theta_i,\theta_j\}=0$$.
The Leibniz rule for the Grassmann number was given as definition, $$\frac{d\theta_i\theta_j}{d\theta}=\frac{d\theta_i}{d\theta}\theta_j-\...
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Grassmannian $\hbox{Gr}(2,\mathbb{C}^5)$ is homeomorphic to $\hbox{Gr}(3,\mathbb{C}^5)$
How to show that $\hbox{Gr}(2,\mathbb{C}^5)$ is homeomorphic to $\hbox{Gr}(3,\mathbb{C}^5)$ by showing that they are given by the same Plücker relations?
I'm trying to understand Grassmannian and ...
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Algebraically independent Plücker relations
I'm trying to find algebraically independent Plücker relations for $Gr(2,\mathbb{C}^5)$ which generate the ideal. How do I find them and how do we prove if a Plücker relation is algebraically ...
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Computing the Stiefel-Whitney classes of the tautological bundles $\gamma^n\rightarrow Gr(n,\mathbb R^\infty)$
I was reading about the Stiefel-Whitney classes construction using Grassmannians on Wikipedia's page.
It seems to me that all the axioms can be deduced once the $k$-th Stiefel-Whitney class of a rank $...
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Find the subspace of $\mathbb{C}^n$ with given Plücker coordinates.
I'm studying Representation theory and I'm trying to understand the Grassmanian and Plücker coordinates. This problem is an exercise from Young Tableaux: With Applications to Representation Theory and ...
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What is the base point of the Stabilizer that forms the subspace $Span${$e_1$,....,$e_k$}
I am preparing for an exam for my linear algebra class and I had this question pop up in my mind.
What is the base point for the isotropy group that is the subspace $Span${$e_1$,....,$e_k$}
This ...
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Prove that the group of invertible matrices acts transitively on Grassmannian space
I have recently learnt about group actions and grassmannian space. So While searching further on the topic about general linear groups acting on different sets, I read something on wikipedia:
$GL$($V$...
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dense subset in Grassmann manifold
for integers $k$ and $n$ with $1\leq k\leq n-1$. Let $G_k(\mathbb{R}^n)$ denote the set of linear subspaces of dimension $k$ in $\mathbb{R}^n$, and let $X_{n,k}=\{A\in M_{n,k}(\mathbb{R})|\mbox{rank}(...
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Does a finite-dimensional Grassmannian classify the subbundles of a trivial vector bundle?
It is known that the universal vector bundle over the infinite-dimensional Grassmannian,
$$
E \longrightarrow Gr_n(\mathbb{R}^{\infty}),
$$
classifies the rank $n$ vector bundles in the sense that any ...
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Identification of Lagrangian Grassmannian with unitary group
$\DeclareMathOperator{\Ker}{Ker}$
Consider the standard symplectic structure on $\mathbb C^{2n}$ induced by the symplectic matrix $J$. We can write: $$\mathbb C^{2n}=\Ker(J-iI)\oplus \Ker(J+iI)$$
This ...
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A metric on the Grassmannian manifold induced by the Riemannian metric
In Wikipedia, they define the Grassmannian manifold by $$Gr(r,n)=O(n)/(O(r)\times O(n-r))$$ where $O(m)$ is the orthogonal group of $m\times m$ matrices. They say that it gives a metric on the ...
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Cohomology ring of grassmannian and Pieri rule
I am learning Schubert variety and I came across a problem to understand a particular detail (I asked the same question on mathoverflow : https://mathoverflow.net/questions/397999/cohomology-ring-of-...
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books on dual projective varieties and grassmannians
I'm currenty writing my bachelor degree thesis on the dual projective variety with focus on the biduality theorem. I'm looking for a book where to find this topics and a book that explains the ...
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Cohomology group of Grassmannian manifold
Here we hope to confirm the cohomology group of Grassmannian manifold, also this manifold behaves as some homogeneous spaces obtained from Lie groups. In particular with the coefficients of mod 2 or ...
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reference for cohomology ring of even orthogonal grassmannian
Even orthogonal Grassmannian $OG(m,2n)$ are the spaces parameterize $m$-dimensianl isotropic subspaces in a vector space $V\simeq \mathbb{C}^{2n}$, with a nondegenerate symmetric bilinear form. It's ...
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Normal Bundle to flag manifold/variety
Let $Fl(k_1,k_2) = \{ (V_1,V_2)\in G_{k_1}(V)\times G_{k_2}(V):\text{ }V_1\subset V_2\}$ be a two step flag manifold over a $n$ - dimensional vector space $V$. I am mostly interested in the case $V =\...
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The ring cohomology of the complex Grassmanian $G_k(V)$
I am trying to understand proposition 23.2. of Bott Tu proof. It wants to prove that $H^{*}(G_k(V))=\frac{\mathbb{R}[c(S),c(Q)]}{(c(S)c(Q)=1)}$.
It claims that:
1.$H^{*}(F(V))=\frac{\mathbb{R}[x_1,\...
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Is the Kempf-Laksov-resolution of a Gorenstein single-condition Schubert variety a blowup?
Let $Gr(k,V)$ be the Grassmann bundle of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ equipped with a full flag $0=E_0\subset E_1 \subset \ldots \subset E_{n-1}\subset E_n=V$. ...
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Easier proof that the Grassmannian is a complex manifold
$G_r(\mathbb C^3,2)$ is the topological space of 2-dimensional complex linear subspaces of $\mathbb C^3$. Prove that $G_r(\mathbb C^3,2)$ is a complex manifold.
I have a solution to this problem, but ...
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Pushforward from Flag variety
Consider a vector space $V$ and some $n,n+1\leqslant\dim V$.
Let $F\subset G(n,V)\times G(n+1,V)$ be the Flag variety (where $G(n,V)$ is the Grassmannian of $n$-dimensional subspaces of $V$),
i.e. the ...
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Reference request: cohomology ring of flag varieties
Just when I started understanding the basics of Schubert calculus and how the cohomology ring of Grassmannians $G(k,n)$ works, I figured I needed a generalization in terms of (partial) flags.
My goal ...
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