The grassmannians tag has no wiki summary.

**2**

votes

**1**answer

107 views

### Rational normal curves on Grassmanians

Consider the Grassmanian $G(k,n)$ ($k\le \frac{n}{2}$) and take its Plucker embedding. Consider now the space of all normal rational curves of degree $k$, contained in the Plucker embedding of the ...

**1**

vote

**0**answers

131 views

### Chern classes of tautological bundle over the Grassmannian G(2,4) [migrated]

I've the following problem: I know how to calculate chern classes of the tautological bundle over the Grassmannian $G=G(2,4)$ using the Shubert calculus. If I am right, the chern character should be
...

**1**

vote

**1**answer

228 views

### Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $LG(2,4)$?

On the paper
http://arxiv.org/abs/1009.1364
(published on Proc. London Math. Soc.) I've found an interesting statement:
The Lie quadric $Q^3$, i.e., the space of all points, lines and
circles ...

**2**

votes

**1**answer

372 views

### Two questions about the grassmannian

There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference.
...

**2**

votes

**2**answers

165 views

### Curve of 3-secant lines

Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.
Consider the variety ...

**1**

vote

**0**answers

108 views

### Grassmannian frames in the Grassmannian

I am new to the Grassmannian. I have read about Grassmannian frames in $\mathbb R^n$. My question is can we define Grassmannian frames in a Grassmannian space $Gr(k,n)$ just like in $\mathbb R^n$? ...

**0**

votes

**0**answers

49 views

### What is the best way to optimize this matrix equation

What is the best way to optimize this Procrustes like formulation:
$\min\quad\|AX-B\|^2_{\rm F} + \|X^Tc\|^2,$
s.t. $X^TX = I$
Here A and B are $n \times p$ matrices and $c$ is a $p \times 1$ ...

**4**

votes

**2**answers

196 views

### Upper bound for the product of Schubert cycles

Let $Gr(c,\infty)$ be the complex grassmannian of $c$-dimensional subspaces of the infinite dimensional complex space. Every finite dimensional grassmannian, $Gr(c,N)$, can be thought as a subspace of ...

**5**

votes

**2**answers

276 views

### Relations between affine Grassmannian and Grassmannian

Let $\mathcal K = k((t))$ be the field of formal Laurent series over $k$, and by $\mathcal O = k[[t]]$ the ring of formal power series over $k$.
Let $G$ be an algebraic group over $k$. The affine ...

**0**

votes

**0**answers

40 views

### Fitting ideals and a Grassmannian construction

Let $L$ be a locally free and finitely presented sheaf over a Noetherian scheme $X$ and
$$ E\overset{\varphi}\to F \to L \to 0$$
a free presentation of $L$, where $E$ and $F$ have finite ranks $n$ ...

**1**

vote

**0**answers

179 views

### Elegant definition for the scheme parametrizing $g_d^r$'s on a curve

Let $X$ be a smooth projective curve over $k=\bar{k}$, and $Pic^d$ the $d$-part of the Picard group of $X$ (isomorphism classes of line bundles of degree $d$ on $X$).
I'd like to define a scheme ...

**1**

vote

**1**answer

125 views

### A calculation over product of Grassmannians

Let $c,d<N$ be integers and consider the product of two Grassmannians $M=Gr(c,N)\times Gr(d,N)$. Define $S\subset M$ to be the set of the pairs $([A_{c\times N}],[B_{d\times N}])$ such that ...

**2**

votes

**0**answers

72 views

### Computing intersection of cycles on the product of Grassmannians/Deligne-Lusztig varieties

My collaborators and I are preparing an interesting manuscript where the computation leads to something related to what we believe to be in the area of Schubert calculus; but none of us knows much ...

**51**

votes

**1**answer

3k views

### What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...

**48**

votes

**1**answer

4k views

### The amplituhedron minus the physics

Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it?
All the descriptions I've so far encountered assume ...

**6**

votes

**1**answer

291 views

### What is known about dynamics on Grassmannians?

I have found myself becoming interested in dynamical systems given by homeomorphisms acting on $G(r,d)$, the space of $r$-dimensional subspaces of $\mathbb{R}^d$. I tried to do a literature search ...

**7**

votes

**2**answers

302 views

### Is this a metric on the Grassmannian Manifold?

Let $m>n$ and consider the Set
$$S_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I_n \}.$$
Does the function $d\colon S_{m,n} \times S_{m,n} \rightarrow \mathbb{R}$ defined by
...

**0**

votes

**1**answer

94 views

### References on complete intersections in Grassmanian

Are there any (there should be) references on complete intersections in Grassmanian? Especially on calculating the cohomolgy of some sheaves naturally associated to the complete intersection. For ...

**1**

vote

**0**answers

263 views

### Are Grassmannians toric varieties? [closed]

Are Grassmannians $G(k,n)$ toric varieties for all possible $k,n$? If they are toric varieties, are there any descriptions for the fans?

**4**

votes

**4**answers

405 views

### Submanifolds in the Grassmannian of n-dimensional subspaces determined by a submanifold in the Grassmannian of l-dimensional subspaces

Let $G_n(V)$ the Grassmannian of $n$-dimensional subspaces of a finite-dimensional $V$, and $l<n$. I've noticed that it is easy to associate a (possibly singular) submanifold $\tilde{M}\subseteq ...

**2**

votes

**0**answers

128 views

### A natural tower of bundles over Grassmannian manifolds

There is a well-known exact sequence of vector bundles
$$
0\to U\to T\to N\to 0
$$
over a Grassmannian manifold $Gr(V,n)$: the fibers over $L\in Gr(V,n)$ are, respectively, $L$ itself, the whole $V$ ...

**1**

vote

**1**answer

431 views

### Canonic identification of the tangent space of the Grassmannian

let $Gr(k,V)$ be the grassmannian of k-dimensional subspaces of the complex vector space $V$ of dimension $n>k$.
I know that, given $K\in Gr(k,V)$, $T_{Gr(k,V),K}\simeq Hom(K,V/K)$, but i want to ...

**10**

votes

**0**answers

233 views

### Embedding of the product of two Grassmannians into a Grassmannian

Consider an embedding $$\Phi: G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})\rightarrow G_k(R^n)$$ of the product of two Grassmannians $G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})$ into $G_k(R^n)$, where ...

**1**

vote

**1**answer

133 views

### Tangent space to positive oriented Grassmannians

Let $L$ be a real vector space of dimension 22 and $q$ a quadratic form on $L$ of signature $(3,19)$.
Let $V\subset L$ be a positive oriented subspace of dimension 2 and $G^{po}(2,L)$ be the ...

**1**

vote

**1**answer

123 views

### Non-(stable)-triviality of the tautological bundles

This is a question I asked at Math.SE but got no answers: http://math.stackexchange.com/q/396217/7110/
The tautological vector bundle $\gamma_k(\mathbb{K}^N)$ over the Grassmann manifold ...

**6**

votes

**2**answers

440 views

### Volume of Gr(2,4)

Hello
I was wondering if anybody can direct me to a paper or a book regarding the volume of $Gr(2,4) $ or generic complex Grassmanian manifolds of order $k$. My own heuristic method seems not to ...

**3**

votes

**0**answers

292 views

### Tropicalization of the Grassmannian

Let $Trop(Gr(m,n))$ denote the tropicalization of the grassmannian $Gr(n,m)$. Let $\phi^m : \mathbb R^{n \choose 2} \rightarrow \mathbb R^{n \choose m}$ such that $X_{i,j} \rightarrow ...

**1**

vote

**1**answer

155 views

### Is this an embedding of $S^{[2]}$?

The intersection of 3 quadrics in $P^5$ is a K3 surface $S$.
There is a natural map $S^{[2]} \to G(1,5)$ well defined everywhere, because a generic K3 doesn't contain any line and this family is ...

**11**

votes

**2**answers

513 views

### What are some triangulations of Grassmannians?

A while ago I heard that there was no known triangulation of the Grassmannian of 3-planes in 6-space.
To believe a statement like that, you have to be a little bit ungenerous about what you mean by ...

**2**

votes

**1**answer

308 views

### Grassmannian of oriented real $k$-planes

The Grassmann manifold $\widetilde{Gr}(k,\Bbb{R}^n)$ of oriented $k$-planes in $\Bbb{R}^n$ is a double cover of the Grassmann manifold $Gr(k,\Bbb{R}^n)$ of non-oriented $k$-planes. We can give ...

**1**

vote

**0**answers

175 views

### Odd-Dimensional Complex Quadrics

It's well known that the even-dimensional complex quadric $Q_{2n}$, defined by the equation $z_1^2+\cdots +z_{2n+2}^2=0$ in complex projective space $CP^{2n+1}$, is diffeomorphic with the Grassmann ...

**3**

votes

**2**answers

238 views

### What is the ideal corresponding to the Plücker embedding?

Let $S$ be a noetherian scheme, $\mathcal{E}$ a quasi-coherent sheaf on $S$ and let $d \in \mathbb{N}$. There is a Plücker embedding $\omega : \mathrm{Grass}_d(\mathcal{E}) \hookrightarrow ...

**4**

votes

**2**answers

295 views

### Impossibility of continuously picking k independent rows from a rank k matrix

Suppose I have an $n\times n$ real (or complex) matrix of rank $k$, and I want to pick $k$ linearly independent rows from it. I want to do this in a continuous fashion as the matrix varies ...

**5**

votes

**1**answer

216 views

### Little puzzle about intersections of Schubert cells.

I was reading chapter 6 in the book of Harris on Algebraic geometry and came to the following puzzle.
It seems to me that every Schubert cell in a Grassmanian is obtaining by cutting the Grassmanian ...

**0**

votes

**0**answers

176 views

### orthogonal VS. lagrangian grassmannian

I want to learn more about orthogonal and lagrangian (or symplectic) grassmannians. What are good references?

**4**

votes

**1**answer

218 views

### On totally nonnegative Grassmannian

I was reading Postnikov's paper [TOTAL POSITIVITY, GRASSMANNiANS, AND NETWORKS][1] when I came across the definition of the totally nonnegative Grassmannian $Gr_{kn}^{tnn} \subset Gr_{kn}$ as the ...

**2**

votes

**2**answers

215 views

### product of two sub-Grassmannians

Let $G(k,n)$ be the Grassmannian of complex $k$-planes in $\mathbb{C}^n$. Then for $k_1+k_2=k$ and $n_1+n_2=n$, $G(k_1,n_1)\times G(k_2,n_2)$ is a submanifold of $G(k,n)$. So the cohomology class of ...

**2**

votes

**1**answer

117 views

### How to characterize the dual of an isotropic hyperplane?

Hi there! I have a very simple question, which requires an expert in multilinear algebra.
$V$ is an $n$-dimensional vector space, and $\omega\in V^\ast\wedge V^\ast$ is a skew-symmetric form on it. ...

**8**

votes

**5**answers

686 views

### Projective spaces as affine varieties

This works over the reals but not over the complex field. Consider the set of all $n\times n$ matrices $A$
such that
1. $A^2=A$
2.$A^T=A$
3. $\mathrm{Trace}(A)=1$
The first condition makes $A$ a ...

**6**

votes

**2**answers

356 views

### Generalized Grassmannians that parameterize the submodules of a module

I'm looking for something like a Grassmannian, but which parameterizes the submodules of a module rather than the subspaces of a vector space. Most specifically, I'm looking for something which ...

**2**

votes

**1**answer

357 views

### Osculating spaces and distributions on (real) Grassmannian manifold

Hello! Recenlty, doing my research, I came across a quite natural construction, and I would like to know more about it. Unfortunately, being not expert neither in Grassmannians nor in Contact ...

**0**

votes

**1**answer

353 views

### Tautological and normal bundles over flag manifolds and jet bundles

Hello! Recently, doing my research on jet bundles, I was led to consider the following construction.
Let $V$ be a real vector space of dimension $n$. Consider the flag manifold $G(V,k,l)$ and the ...

**8**

votes

**4**answers

782 views

### Riemannian metric on a flag variety

$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = ...

**3**

votes

**0**answers

175 views

### Extending intersection bundles

Let $X$ be the product $Gr_i(V)\times Gr_j(V)$ of two Grassmannians where $V$ is a complex vector space of dimension $d$. There is an open $U\subset X$ formed by all those $(V',V'')\in X$ such that ...

**5**

votes

**1**answer

496 views

### Embedding $G(2,n)$ into $G(k,n)$

Let
$$M=\begin{pmatrix}
u_1 & u_2 & \ldots & u_n \\
v_1 & v_2 & \ldots & v_n \\
\end{pmatrix}$$
be a $2 \times n$ matrix. Define $\nu(M)$ to be the $k \times n$ matrix
...

**6**

votes

**2**answers

333 views

### Homotopy equivalence between the Grassmannian Gr_{n,m} and Gr_n \times Gr_m.

The following assertion appears in a paper I am reading, and I can't seem to verify it.
Let $\text{Gr}_{n,m}$ denote the set of pairs $(V,W)$ where $V$ and $W$ are as follows.
$V$ is an ...

**2**

votes

**2**answers

496 views

### Smoothness of hypersurfaces in Grassmannians

I have a general question, and then the specific version of that question I need for research. All vector spaces over $\mathbb{C}$.
Grassmanians of planes
The $(2,n)$-Grassmannian, denoted ...