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2
votes
1answer
160 views

Schubert calculus and Pieri's formula

In the lecture notes Grassmannians: the first example of a moduli space. MIT Open Course Ware. page 7: Are there any formal publications (books/papers) where I can find the formula?
6
votes
1answer
144 views

Steenrod operations on cohomology of grassmannians

Let $G_k(\mathbb{R}^n)$, $n\geq k$ and $G_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of ...
5
votes
1answer
253 views

Cohomology of $G_3(\mathbb{R}^5)$

This is in some sense a specialization of the question integral or rational cohomology of real grassmannians. Let $G_3(\mathbb{R}^5)$ denote the real Grassmannian of (unoriented) $3$-planes in ...
2
votes
1answer
101 views

Are cones over Grassmannianns of lines local complete intersections?

Let $X_d^N\subset\mathbb{P}^N$ be a cone over the Grassmannian of lines $\mathbb{G}(1,d)\subset\mathbb{P}^{d(d+1)/2-1}\subset\mathbb{P}^N$ with vertex a linear space $L\subset\mathbb{P}^N$ of ...
1
vote
2answers
130 views

Transformations that leave the Plucker embedding of G(2,4) invariant

I am interested in a group of transformations that leave the Plucker embedding of complex Grassmannian $G(2,4)$ into $CP^5$ given by ...
4
votes
0answers
259 views

Is there a well-known tautological bundle over $\mathbb{P}(\Lambda^nV)$?

Let $V$ be a vector space of dimension $>n$, and define the subset $$ K:=\{ ([\omega],v)\mid v\wedge\omega=0 \}\subset\mathbb{P}(\Lambda^nV)\times V\, . $$ Denote also by $\pi:K\longrightarrow ...
11
votes
0answers
197 views

q-Catalan numbers from Grassmannians

In this question by $q$-Catalan numbers I mean the $q$-analog given by the formula $\frac{1}{[n+1]_q}\left[{2n\atop n}\right]_q$. The polynomial $\left[{2n\atop n}\right]_q$ represents the class of ...
1
vote
0answers
98 views

Metric(s) on Grassmann Manifold and Plucker Embedding

I'm working on a numerical optimization problem that naturally lives on the Grassmann Manifold Gr$_N(\mathbb{C^M})$, however the objective function is defined on the alternating algebra given by the ...
1
vote
0answers
58 views

Schubert Calculus for Quaternion-Kähler Manifolds

The cohomology ring of general Grassmannians have very nice presentations in terms of Young diagram and the rules of Littlewood-Richardson. This is called {\em Schubert calculus}. The Grassmannian of ...
1
vote
1answer
164 views

Homotopy type of certain maps on complex grassmanian

$G(k,n)$ is the complex grassmanian which is homeomorphic to the space of projections in $M_{n}(\mathbb{C})$ with trace $k$. So we can Identify $G(k,n)$ with $$\{A\in M_{n}(\mathbb{C})\mid ...
0
votes
0answers
82 views

rational cohomology of finite dimensional real grassmannian

Let $G_k(R^n)$, $n>k$, be the finite dimensional real grassmannian. What is the rational cohomology algebra $H^*(G_k(R^n);Q)$? I have searched out that $H^*(BO_k;Q)=Q[p_1,p_2,...,p_[k/2]]$ is the ...
2
votes
1answer
146 views

rational cohomology of finite real grassmannian

Let $p_j$ to be the $j$-th Pontryagin class of the universal $n$-plane bundle $E_n(\mathbb{R}^\infty)\to G_n(\mathbb{R}^\infty)$. Then according to Theorem 1.6, The Cohomology of BSO n and BO n with ...
2
votes
1answer
204 views

Presentation of the tautological bundle of the Grassmannian

Consider a Grassmannian $G=Gr(r,n)$ embedded in projective space $P^n$ by its Plucker embedding. Is there a way of writing down a presentation of the tautological bundle of $G$, as a module over the ...
0
votes
0answers
81 views

Equivalence of Kahler structures of based loop group and its Grassmannian model

In Pressley-Segal's Loop Groups, we have the following spaces equipped with Kahler structures. Let $G$ be a compact, connected, (simply connected) group with Lie algebra $\mathfrak g$. Let ...
1
vote
2answers
332 views

integral or rational cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...
2
votes
0answers
44 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
109 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
86 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 ...
21
votes
7answers
2k 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
270 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 ...
3
votes
1answer
122 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
301 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
439 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
204 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
125 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 $\mathbb R^n$? ...
4
votes
2answers
206 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
395 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
52 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
188 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
139 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
81 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 ...
57
votes
3answers
4k 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 ...
53
votes
2answers
6k 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
328 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
453 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
103 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
368 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
514 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
143 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 $V$ ...
2
votes
1answer
772 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 ...
11
votes
0answers
272 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
144 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 ...
2
votes
1answer
134 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
536 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
319 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
167 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 ...
13
votes
2answers
615 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
408 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
223 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
270 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 ...