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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Fundamental Group of complex Grassmannian

How can we compute the fundamental group of complex Grassmannian Gr(k,n) using Van Kampen Theorem only? I know it will be simply connected, so we better find two simply connected open sets which cover ...
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Are dual numbers related to dual spaces? [closed]

I recently came across the notion of a dual number and am curious if there is any direct relation to a dual space. I can't really find anything to answer this, though the notion of a Grassmann number (...
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The set of lines through a fixed point in $\mathbb{P}^3$

I am struggling to solve this question. I have to find the set of lines through a fixed point in $\mathbb{P}^3$ using $Gr(2,4)$. How can I show that this set is inside $Gr(2,4)$ and a subset in $\...
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Equivalent definitions Stiefel - Whitney / Chern - classes

There are some details I still don't understand about the definition of Stiefel - Whitney / Chern - classes. Let $\gamma_n^1$ be the tautological line bundle over the real or complex projective space $...
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Given a vector bundle over a compact space, how can we determine the smallest Grassmannian classifying it?

It is a standard result that for $\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$, the Grassmannian $G_n(\mathbb{F}^{\infty})$ is a homotopical classifying space for $n$-plane bundles over any ...
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Lines on grassmannian

Given a projective space $\mathbb{P}^n(\mathbb{C})$, I can consider the Grasmannian of lines $G(2,n+1)$, which has a structure of projective variety inside $\mathbb{P}^N$, where $N=\binom{2+n+1}{2}-1$...
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How far can an $N$-fermion wavefunction be from the nearest Slater determinant?

Consider a system of $N$ non-relativistic spin-$0$ fermions in $D$-dimensional space, all of the same species. The wavefunction of such a system can be represented by a function $\psi(\mathbf{x}_1,...,...
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Schubert calculus

Let $X = Gr(2,4)$ the complex Grassmannian of $2$-planes in $V = \Bbb C^4$ and $S$ the tautological bundle, $Q$ the quotient bundle. The cohomology ring is generated by $c_1(S), c_2(S)$ with relations ...
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Describe $\mathcal{N}_{G(\mathbb{P}^1,\mathbb{P}^k)\mid G(\mathbb{P}^1,\mathbb{P}^n)}$

Consider $1 \leq k < n$ positive integers, and denote by $G(\mathbb{P}^k,\mathbb{P}^n)$ the Grassmannian of $\mathbb{P}^k$'s in $\mathbb{P}^n$. It is well known $G(\mathbb{P}^k,\mathbb{P}^n)$ ...
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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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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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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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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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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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$C^m\to C^{m-1}$ 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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Grassmannian as a quotient of orthogonal or general linear group

I'm trying to understand some different ways to construct the Grassmannian of a real vector space, but I'm having trouble getting some of the notation and definitions. One definition that I often see ...
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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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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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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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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 ...
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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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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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Trying to understand some basic facts about tangent space of Grassmannian.

I am reading Harris's 'Algebraic Geometry: A first course'. I am trying to understand its identification of the tangent space to a Grassmannian. Let $G(k,n)$ be the Grassmannian of $k$-planes in $K^{n}...
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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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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 ...
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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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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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