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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Grassman variable and Grassmannian?
Grassman variables are anticommuting number or supernumber, is an element of the exterior algebra over the complex numbers.
Grassmannian $Gr(k, V)$ is a space that parameterizes all $k$-dimensional ...
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Grassmannian is a manifold in a specific case the $2$-planes in $\mathbb{R}^4$
I want to show that Grassmannian is a manifold in a specific case the $2$-planes in $\mathbb{R}^4$.
I'm in the following context:
$G(2,4)$ are the $2$-planes in $\mathbb{R}^4$ that we can identify ...
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Lagrangian grassmannian of all lagrangian subspaces in $\mathbb{R}^n\times \mathbb{R}^n$ can be identified with $U(n,\mathbb{C})/O(n,\mathbb{R})$
I've been trying to proof this using $U(n,\mathbb{C})$ action over all lagrangian subspaces of $\mathbb{R}^n\times \mathbb{R}^n$ but it didn't work. I mean, I got stuck and I didn't know what else to ...
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Grassmanian is a manifold (Hirsch)
I am trying to do an exercise from Hirsch's Book , Differential Topology, that basically wants me to prove that $G_{n,k}$ is a manifold.
The Grassmanian manifold $G_{n,k}$ of $k$-dimensional ...
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Ring of functions on the Grassmannian
Background: Let $k$ be a commutative ring. In general for a $k$-functor $X : \textbf{Alg}_k \to \textbf{Set}$ we define the ring of functions on $X$ to be $\textbf{Nat}(X , \mathbb{A}^1)$ with the ...
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Show that the $GL(n,\mathbb R)/P_k$ is isomorphic to the $GL(n,\mathbb R)$-set grassmannian.
Let $r<n$ be two positive integers and $G=GL(n,\mathbb{R}).$ If $Gr(k,\mathbb{R}^n)$ is the set of all $k$-subspaces, then show that the $G$-sets $Gr(k,\mathbb{R}^n)$ and $G/P_k$ is isomorphic, ...
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Why are rational functions invariant under symmetric linear group generated by Plucker coordinates
I am working on an exercise and not sure where to start.
Let $K=(\mathbb{C}^2)^{n+3}$. The special linear group $SL_2$ acts naturally on each $\mathbb{C}^2$ and hence on $K$. Let $R$ be the field of ...
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Are linear subspaces of dimension $d$ that exclude a variety of dimension $n-d-1$ an open subset of the Grassmannian?
Suppose I'm working in $\mathbb{P}^n$ and I have an irreducible algebraic variety $X$ of dimension $n-d-1$. In the Grassmannian of dimension $d$, can I always find an open set $U$ such that none of ...
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Oriented Grassmann is a $2$-sheeted covering space of Grassmann
Let $G_n(\Bbb R^k)$ denote the Grassmann manifold (consisting of all $n$-planes in $\Bbb R^k$), and let $\tilde{G}_n(\Bbb R^k)$ denote the oriented Grassmann manifold, consisting of all oriented $n$-...
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What vector bundles are tangent bundles of smooth manifolds?
Given a smooth manifold $M$, we can naturally associate to it a vector bundle ${\rm T}M\rightarrow M$ called the tangent bundle of $M$. This operation induces a functor $\rm T:\rm Diff\rightarrow\rm ...
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Grassmannian $Gr_{\mathbb{R}}(r, n)$ and $Gr_{\mathbb{C}}(r, n)$ as homogeneous space?
It is known that Grassmannian as a homogeneous space https://en.wikipedia.org/wiki/Grassmannian#The_Grassmannian_as_a_homogeneous_space
gives that:
$$
{{Gr_{\mathbb{R}}(r, n) {{=}} O(n)/(O(r) \times O(...
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Gluing Construction of the Grassmanian in Eisenbud/Harris
On page 119 of Eisenbud and Harris' "The Geometry of Schemes," they construct the Grassmanian by gluing. We start by identifying $k$-dimensional subspaces of an $n$-dimensional space $K^n$ ...
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Differential of a canonical map
While studying about curvatures I came up with the following but was unable to work it out fully. Let $M \subset \mathbb{R^n}$ be an embedded submanifold of dimension $k$. Then there is a natural (...
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Derivative of Gauss map is the second fundamental form
I have been messing around with Grassmannians lately.
Let $M^k\subseteq \Bbb R^n$ be an embedded submanifold equipped with the induced Riemannian metric, and consider the Gauss map $G\colon M \to {\rm ...
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Understanding Grassmannian as $\text{Gr}(k,n) \cong \text{GL}(k,k) \ \backslash \ \text{Mat}^*_{\mathbb{R}}(k,n)$.
The Grassmannian $\text{Gr}(k,n)$ can be described as the quotient
$$\text{Gr}(k,n) \cong \text{GL}(k,k) \ \backslash \
\text{Mat}^*_{\mathbb{R}}(k,n) $$
where $\text{Mat}^*_{\mathbb{R}}(k,n)$ is ...
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If and how is this related to Grassmanians?
Given a $n$ dimensional vector space over the finite field $F_q$, called $V(F_q)$, and a set of $M$ vectors $\vec c_m=(c_0,c_1,...c_{n-1})^T$ that fulfill $(\sum c_k )=0$. Further the complete set of $...
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Divisors on subvarieties of the Grassmannian
A statement that I find sometimes in different forms and it puzzles me.
I call $\mathbb G(1,N)$ the Grassmannian, seen as a submanifold of $\mathbb{P}(\bigwedge^2 \mathbb{C}^{N+1})$, and $H^l$ a ...
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Automorphisms of a manifold inside a Grassmannian
Given a smooth manifold $X$ with a very-ample line bundle $L$, we have an associated embedding
$$ \phi_{|L|}:X \to \mathbb{P}(H^0(X,L)^\vee) .$$
We know that if an automorphism $\alpha:X\to X$ leaves ...
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How is $\mathbb{F}_2^4$ related to an $8$ element set?
I am trying to understand the part of this answer explaining why $A_8\cong\mathrm{PSL}_4(\mathbb{F}_2)$.
Let $|X|=8$. We can form the free vector space $\mathbb{F}_2X=\mathbb{F}_2^8$ with the usual ...
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Induction Proof for Computing Cohomology Ring of the Finite Grassmannian
I'm working on problem 7B of Milnor/Stasheff:
Show that the cohomology algebra $H^*\left (G_n\left
(\mathbb{R}^{n+k}\right ) \right )$ over $\mathbb{Z}/2$ is generated
by the Stiefel-Whitney ...
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Is there a reasonable notion of a totally real Grassmannian?
Suppose we are given a $2n$-dimensional real vector space $V$, along with a complex structure $J:V\to V$. For $k\le n$, a $k$-subspace $W\subset V$ is called $J$-totally real if $W\cap JW = 0$. ...
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If $U \subset M_{n,k}(\mathbb{R})$ is open, and $g \in GL_k(\mathbb{R})$, then how to show that $Ug$ is open?
This question is from my course of smooth manifolds.
Let $G(k,n)$ denote $\{k\text{-dimensional vector space in } \mathbb{R}^n\}$,
which is equal to $\{n \times k \text{ matrix of rank }= k\} \big/ \...
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Lines on a quintic del Pezzo 3-fold of degree 5
The quintic del Pezzo $3$-fold $V(5)$ of degree $5$ is defined as the intersection of $Grass(2,5)$ and a codimension $3$ linear subspace. I would like to show the following using only elementary ...
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Transversality of Flags
I'm struggling to understand the transversality of flags:
The statement "any general pair of distinct flags $\mathbb{F}_{1}$
and $\mathbb{F}_{2}$
can be mapped by a suitable element of GL(n) to a ...
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non totally decomposability of vectors in Grassmannian
In "Algebraic geometry: a first course", by Harris, Grassmannian is described, under the Plucker embedding, as the locus of totally decomposable vectors in the projectivization of the exterior power $\...
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Grassmannians and $\mathrm{GL}(n,\mathbb{R})$
Let $\mathrm{Gr}_n$ denote the infinite real Grassmannian of $n$-planes in $\mathbb{R}^\infty$. This is a classifying space for real vector bundles, in the sense that (for paracompact $B$)
$$ [B, \...
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Compactness of the set of rank K projectors
I hope you could give a hint for proving that the following set is compact $(k<n)$:
$X=\left\{A\in \mathbb{R}^{n\times n}:A=A^{t},A^{2}=A,rank(A)=k\right\}$
I can proof that $X$ is bounded(not so ...
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About embeddings of real grassmannians
I am dealing with the embedding of real grassmannians $G(n,k)$ on $\mathbb{R}^{n^2}$ via the map associating to each vector space the projection matrix on it in the canonical base of $\mathbb{R}^n$. ...
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Grassmannians: Varieties swept out by linear spaces (Eisenbud & Harris: 3264 and All That)
I have a couple of questions on statements from Harris' and Eisenbuds's lecture "3264 and All That" at page 145, Section 4.2.3: Varieties swept out by linear spaces.
The content of 4.2.3 answers ...
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Conics in Grassmannians
Consider the complex projective space $\mathbb{P}^4$ and the Grassmannian $\mathbb{G}(1:\mathbb{P}^4)$ of lines in it, seen as a projective manifold through plucker embedding.
Take $l_1,l_2\subset \...
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Is the map assigning $x$ to its tangent space smooth?
I'm having some difficulties understanding smooth maps between manifolds, and in particular I would like to know if the map $x \mapsto T_{x}M$ is a smooth map $M\to Gr(k,n)$? Here $x \in M$, where $M$ ...
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Generically transversally intersecting Schubert cycles
I have a question about the the proof of Pieri's formula from Harris' and Eisenbuds's lecture "3264 and All That"on page 146. Before the proof we use this terminology (see page 139):
let $G=G(k,V)$...
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1answer
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Is there any entity that possess information of magnitude,direction,starting and ending points?
Is there any entity similar to vectors but also possess the starting and ending points ?
For instance, consider a plane $z = 4$, Suppose I want a vector starting from A$(0,0,4)$ and ending at B$(0,1,...
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Properties of a general element of the intersection of two Schubert Cycles
We have the following Lemma:
Lemma
Let $\Sigma_a(\mathcal{V}),\Sigma_b(\mathcal{W})$ be two Schubert cycle defined relative to transverse flags $\mathcal{V}$ and $\mathcal{W}$. If $\Lambda \in \...
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Basis theorem for the Grassamannian [duplicate]
Does anyone know where I can find a good proof for the basis theorem of the cohomology ring of the Grassmannian, or give me a sketch of the proof?
I'm already familiar with basic Schubert Calculus.
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which mehod can easily design index of Grassmannian and its k and n or any function for this in Macaulay2 and how to convert poset to this index?
how to know k and n and its index of Grassmannian?
which mehod can easily design index of Grassmannian and its k and n or any library or function for this in Macaulay2?
is there any library or ...
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Restricted Isometry Property for Random Orthogonal Projections
The following is from Roman Vershynin's: High-Dimensional Probability: An Introduction with Applications in Data Science.
Let $P$ be the orthogonal projection in $\mathbb{R}^n$ onto an $m-$...
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Schubert cycles that intersect generically transversely.
Let $\mathcal{V}= 0 \subset V_1 \subset \cdots \subset V_{n-1}\subset V_n=V$, $\mathcal{W}=0 \subset W_1 \subset \cdots \subset W_{n-1} \subset W_n=W$ be two flags. We say that $\mathcal{V}$ and $\...
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Cohomology ring of Gr(2, 5)
Does anyone know how to prove that the grassmannian G(2, 5) is generated by the Schubert cycles or some source where I can find a proof?
Thanks!
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Proving that grassmannians are smooth manifolds
I'm trying to show that real grassmannians $G(k, n)$ are smooth manifolds of dimension $k(n-k)$.
The problem is set in this way:
Identify the set of all real matrices with $n$ rows and $k$ columns ...
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A simple question on Grassmann number
Let $(z,\bar{z})$ be complex coordinates in 2D plane, and $\theta$ be Grassmann number, I wonder wether the below equation is right or not. Consider two points denoted 1 and 2
\begin{equation}
\frac{1}...
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An open subset of a special type of Grassmannian obtained from a local ring
Let $(R, \mathfrak m)$ be a Noetherian local ring such that the residue field $k=R/\mathfrak m$ is algebraically closed.
For a fixed integer $t \ge 1$ , $V_t :=\mathfrak m^t/\mathfrak m^{t+1} $ is a ...
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Questions on grassmannians from Harris' and Griffith's principles of algebraic geometry
I have a couple of questions about the explanations on the Grassmannian from Harris' and Griffith's "Principles of algebraic geometry" on page 194/195. We consider for $k \le n$ the $k$-th ...
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$C^m\to C^{m-1}$ projection map projecting out last factor inducing $G_n(C^{m-1})\to G_n(C^m)$?
Assume $m>n$. Consider projection map $\pi: C^m\to C^{m-1}$ by $(z_1,\dots, z_m)\to (z_1,\dots, z_{m-1})$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$ and endow the topology as a ...
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Let $h,h'$ be hermitian metric over vector space $V$, then grassmanian $G_n(V_h)\to G_n(V_{h'})$ is always continuous?
Let $V$ be a finite dimensional vector spaces over complex number and choose hermitian metrics $h,h'$ over $V$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$. Since $V$ has 2 metrics, ...
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Grassmannian is homogeneous, isotropic, and symmetric
I'm trying to prove the Grassmann manifold $\mathrm G_k(\mathbb R^n)$ of $k$-dimensional linear subspaces of $\mathbb R^n$ is isotropic and symmetric.
By "isotropic" I specifically that for every ...
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1answer
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Is the Grassmannian a unique algebraic variety?
I'm fairly new to algebraic geometry, so this is may just be fairly simple question about when two varieties are "the same" variety.
In this question, I noted that the Grassmannian can be expressed ...
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1answer
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What is the way to write a manifold of manifolds?
I want to write this in mathematical notation: "Let us represent a ball, $B_3$, with a metric $g$ as a point on manifold. Let $M$ be the (infinite dimensional) manifold formed from every ball with all ...
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What, exactly, is Schubert's symbolic calculus?
Most modern treatments of the Schubert calculus typically write about the cohomology ring of the Grassmannian. They also write, almost as an afterthought, that this is derived from Schubert's "...
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Stratify Grassmannian based on dimension of intersection
Consider a vector space $V$ of dimension $n$ and an automorphism $A:V\rightarrow V$. Using this we can define, $\Sigma_i = \{W\in Gr_k V| \dim(W\cap AW) = i\}$ for $i=0,\ldots, k$.
My question is ...
