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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Does the orthogonal complement determine the inner product up to scaling?
Let $V$ be a real $n$-dimensional vector space, and let $g,h$ be two inner products on $V$. Fix some $1\le k\le n-1$, and denote by $\text{Gr}_k(V)$ the Grassmannian of $k$-dimensional subspaces of $V$...
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Intersection of lines in $ \mathbb{P}^3 $ can be given as zero locus of homogeneous linear polynomial in $ \mathbb{P}^5 $
I'm currently stuck at Exercise 8.19 b) in the notes to Algebraic Geometry from Gathmann
Let $L \subset \mathbb{P}^3$ be an arbitrary line. Show that the set of lines in $\mathbb{P}^3$ that ...
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1answer
20 views
Meaning of stable $CP^2$
I came across the following phrase in arXiv:1903.08904
....in order to have a stable $CP^2$ , i.e., one in which all the
automorphism group is fixed...
Can anyone explain to me what one means by ...
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Functional differentiation with Grassmann variables
I want to calculate:
$$\frac{\partial}{\partial (x^\rho+i\eta\, \psi^\rho)}[f_\mu(x)+i\eta\, \psi^\nu\partial_\nu f_\mu(x)]$$
where $x^\mu(\tau)$, $\psi^\mu(\tau)$ are commuting and anti-commuting ...
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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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Which subspaces of exterior power have decomposable bases?
Let $V$ be a real $n$-dimensional vector space, and let $1<k<n,r>1$. I wonder:
Is there a way to characterise which $r$-dimensional subspaces of the exterior power $\bigwedge^k V$ have ...
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Grassmannians as Gelfand Pairs
Why are $(O(n), O(k) \times O(n-k))$ and $(U(n), U(k) \times U(n-k))$ (corresponding to the real and complex Grassmann manifolds) symmetric Gelfand Pairs? Is this true for $(Sp(n), Sp(k) \times Sp(n-...
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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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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 $\bigwedge^k V$....
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If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ is a complex power, is it a real power up to a sign?
Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$.
Does there exist $M \in \...
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1answer
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Projecting a projective variety away from a linear subspace
I read in Harris' book at page 148, proposition 11.37 and got slightly confused regarding the argument.
Harris mentions a projective space $\mathbb{P}^{2n+1}$ and a linear subspace $\mathbb{P}^{n}$ ...
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Drawing random subspaces from Grassmannian with uniform probability
Consider the Grassmannian manifold $G(M, N)$ of $M$-dimensional subspaces in $R^N$. I want to approximate (stochastically) an integral of the form
$$
\int_{G(M, N)} f(v) \, dv,
$$
where $f : G(M, N) \...
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Isotropy/little group of $O(n)$
I'm trying to prove that the little group of $O(n)$ acting on a $k$-dimensional subspace of $\mathbb{R}^n$, call it $V$, is $O(k)\times O(n - k)$ due to the Grassmann manifold is isomorphic to $O(n)/(...
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1answer
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Real Grassmann manifold and orthonormal groups
I'm trying to prove that the Grassmann manifold
$$G_k(\mathbb{R}^n) = \{E = {\rm {\it k} - dimensional\ subspace\ of\ } \mathbb{R}^n\}$$
is equivalent to:
$$G_k(\mathbb{R}^n) = \frac{O(n)}{O(k)\...
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1answer
67 views
Straight things on Grassmanians
I am new to Grassmanian, so this question may be too easy. But I didn't find it in any books I know.
When we talk about Grassmanian, it should not be only a manifold or a variety. At least I think we ...
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59 views
General statement about how many lines in Euclidean space will determine a line
It is easy to see that in $3$-dimensional Euclidean space, given $4$ lines in general position, there exists precisely one line who intersects with each of the $4$ lines. We call the $4$ lines ...
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1answer
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Compute the degree of a particular morphism
Let $\mathbb P^n=\mathbb{CP^n}$ be complex projective space. Let $H^0(\mathcal O_{\mathbb P^n}(d))$ be the group of homogeneous polynomials of degree $d$, and denote its dimension by $N(d)$. Consider ...
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Geometric intuition of the dimension of Grassmannians and flag manfolds [duplicate]
I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{...
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Jet prolongation of a distribution on a manifold
I'm trying to work with the first jet prolongation of a $k$-distribution on a manifold $M$ of dimension $n$. My intuition is to consider the Grassmann bundle $X=Gr_k(TM)\to M$ and look at the first ...
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Poincare duality pairing matrix Grassmannian
I'm trying to prove that for $i+j = 2k(n-k)$ and $i = 2s$, $j = 2l$, ($i \neq j$, this is easy) the following Poincaré duality pairing holds:
$$
H^i(\mathbb{G}(k, n)) \times H^{j}(\mathbb{G}(k, n)) \...
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1answer
64 views
The Grassmannian has a non-orientable tangent bundle (in a certain sense)
Let $X$ be a smooth scheme of dimension $r$. Given a rank $r$ vector bundle $\pi: E\to X$, we say that $E$ is orientable if there is a line bundle $L$ on $X$ with an isomorphism $L^{\otimes 2} \cong \...
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1answer
45 views
Relating pullbacks of tautological bundles
Let $X$ be a projective variety and let $\mathcal{E}$ be a vector bundle of rank $r$ on $X$ which is generated by its global sections $V=\Gamma(X,\mathcal{E})$. Recall that this gives us a map $f:X\to\...
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1answer
111 views
How to optimize objective in the Grassmann manifold?
For Stiefel manifold, it contains all the orthogonal column matrices
$$St(d,M) = \{X \in R^{M \times d} | X^TX = I\}$$
For Grassmann manifold, it is
$$Gr(d,M) = \{col(X), X \in R^{M \times d}\}$$
...
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1answer
64 views
How to optimization the $f(X)= \| A - XX^T \|_F^2 + \| X \|_F^2 $ on Grassmann manifold
$$\begin{array}{ll} \text{minimize} & f(X)= \| A - XX^T \|_F^2 + \| X \|_F^2\\ \text{subject to} & X \in Gr(d,N)
\end{array}$$
where $Gr(d,N)$ means the Grassmann Manifold; $\|\cdot\|_F$ ...
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Every Schubert cycle a Chern class?
Consider the Grassmann variety $\mathbb{G}(k,n)$ and its Chow ring $A$. It is known that the classes of Schubert cycles form a $\mathbb{Z}$ basis of $A$. Is it known which of these Schubert cycles can ...
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1answer
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Schubert class in the Grassmannian G(3,6)
How to compute the Schubert class $\sigma$$^2$$_2$$_1$ in the Grassmannian G(3,6)?
I remember the result is $\sigma$$_3$$_3$ + 2$\sigma$$_3$$_2$$_1$ + $\sigma$$_2$$_2$$_2$.
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1answer
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Grassmannian manifold and corresponding vector field
I am assuming the definition of Grassmannian is known.
Reference is Vector bundles and K-theory page no 28.
I am trying to prove that the map $p:E_n(\mathbb{R}^k)\rightarrow G(\mathbb{R}^k)$ is a ...
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1answer
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Plücker embedding - Two definitions
While reading several papers on the topic of the Grassmannian, I cam about two definitions of the Plücker embedding.
One given as
$$ \varphi: \mathbb{A}^{n \cdot d} \rightarrow \mathbb{P}^{\binom{n}{d}...
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1answer
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Affine cone over the Grassmannian
I'm currently working on understanding the affine cone over the Grassmannian, which according to my paper is given by
$$\text{Spec}(K[p_{ij}^{\pm} : ij \in \binom{\lbrack n \rbrack}{2} ] / I_{2,n}).$$...
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Chern classes of tangent bundle over the Grassmannian G(2,4)
What are the Chern classes of the tangent bundle $\tau_G$ of the Grassmannian $G=G(2,4)$ of lines in $\mathbb{P}^3$? This is Exercise 5.37 on page 191 of 3264 & All That by Eisenbud and Harris.
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1answer
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Alternating multilinear function - understanding of specific proof
I was trying to understand a proof that the Plücker coordinates satisfy the Plücker relations. However, I assume that my problems come from a misunderstanding/lack of knowledge about multilinear ...
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Is there a moduli space $\mathcal{M}$ of the closed, $n$-dimensional, simply connected manifolds?
Is there a moduli space $\mathcal{M}$ of the closed, $n$-dimensional, simply connected manifolds with the following properties:
(i) Two abstract simply connected closed manifolds $M, N\in\mathcal{...
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1answer
65 views
Does $Graff(n,\mathbb{R}^{\infty})$ generate all $n$-dimensional closed Riemannian manifolds $M$?
How does one generate all possible $n$-dimensional simply connected closed Riemannian manifolds $M$ from the affine Grassmannian $Graff(n,V)$? Would $Graff(n,\mathbb{R}^{\infty})$ suffice? (It seems ...
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1answer
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BO(-) example in Weiss Calculus
I'nm reading Orthogonal Calculus by Michael Weiss, and trying to understand example 2.7, concerning the derivatives of the functor $BO$, which sends a (finite dimensional) inner product space to the ...
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introduction on Grassmannian on matrix to show $\dim G_r(k,n)=n(n-k)$ [duplicate]
wiki says
In mathematics, the Grassmannian $G_r(k, V)$ is a space which
parametrizes all $k$-dimensional linear subspaces of the
$n$-dimensional vector space $V$. For example, the Grassmannian
...
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Do we need $k = \bar{k}$ for the Grassmanian over $k$ to be an integral, regular projective scheme?
Let $n \geq d \geq 1$ be integers, and $G' := \text{Grass}_{d,n}$ the Grassmanian functor given by
$$
G'(S) := \{\mathscr{U} \subset \mathscr{O}_S^n \mid \mathscr{O}_S^n / \mathscr{U} \text{ is a ...
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On charts of Grassmanian manifolds
Suppose that $V$ is a $K-$vector space. For each positive integer $k$ the set $G_k(V )$ := {$l ⊂ V$| $l$ is a $k$-dimensional linear subspace} is called the Grassmann manifold of $k$-planes in $V$ . ...
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Good Introductory Sources for Grassmannian, Flag, and Stiefel Manifolds
I am looking to gain a deeper understanding of the Grassman, Stiefel, and Flag manifolds but finding good introductory sources so far has eluded me.
I would prefer sources which have:
(1) Concrete ...
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table Betti numbers for real Grassmannians
I am looking for a table of Betti numbers for real oriented and not oriented Grassmannians. Is there some references to get this ?
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1answer
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Grassmann manifold (example Abraham, Marsden, Ratiu book)
I am reading, the book Manifolds, Tensor analysis of Abraham, Marsden and Ratiu. In particular, there are several points that I do not understand about a Grassmann manifold. The example starts like ...
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1answer
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How a Grassmanian Parameterizes a Vector Space
Beginner question. From my understanding roughly, the Grassmannian $Gr(k, V)$ is a space which parametrizes all k-dimensional linear subspaces of the n-dimensional vector space $V$. A vector space is ...
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326 views
The Plucker relations are sufficient
Consider the Grassmannian of codimension-$d$ subspaces of a given vector space $E$ (over an arbitrary field), which I will define as
$$ \operatorname{Gr}^d(E) = \{\text{linear surjections } \sigma: E \...
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1answer
138 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 ...
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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.
...
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1answer
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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{...
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1answer
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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,...
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1answer
61 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 ...
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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]=...
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0answers
21 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 ...
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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}\!\...
