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2
votes
0answers
34 views

Is the product of sine principal angles a semi-metric on Grassmannian?

Let's say we have two $d$ dimensional subspaces with principal angles $\theta_1, \dots, \theta_d$. Then, if $U,V$ are the orthonormal bases for these two subspaces, the singular values of $U'V$ are ...
1
vote
0answers
100 views

(The Homotopy type of the) lifting of homeomorphism of Grassmanian

For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space ...
1
vote
0answers
63 views

Pullback of the tautological vector bundle and the nubmer of trivializations

I've heard about the followign result: for each two natural numbers $d,n \in \mathbb{N}$ one can find a number $k \in \mathbb{N}$ with the following property: for each CW-complex $X$ with $\dim X \leq ...
20
votes
7answers
1k views

Conceptual algebraic proof that Grassmannian is closed in Plucker embedding

I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...
1
vote
2answers
159 views

Plucker embedding and tautological/universal quotient bundle

Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the ...
2
votes
1answer
115 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
1answer
255 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
1answer
394 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
2answers
177 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
0answers
113 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
0answers
51 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
2answers
199 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
2answers
312 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
0answers
42 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
0answers
185 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
1answer
128 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
0answers
75 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 ...
52
votes
1answer
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 ...
50
votes
1answer
5k 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
1answer
305 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
2answers
372 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
1answer
96 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
0answers
293 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
4answers
424 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
0answers
134 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
1answer
503 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
0answers
246 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
1answer
137 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
1answer
125 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
2answers
460 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
0answers
301 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
1answer
156 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
2answers
542 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 ...
3
votes
1answer
341 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
0answers
187 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
2answers
246 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
2answers
296 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
1answer
218 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 ...
4
votes
1answer
238 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
2answers
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
1answer
120 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
5answers
712 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
2answers
367 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
1answer
361 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 ...
1
vote
1answer
403 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
4answers
828 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
0answers
178 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
1answer
503 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
2answers
341 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
2answers
522 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 ...