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4
votes
1answer
75 views

$E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?

Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
7
votes
1answer
100 views

$\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?

Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...
0
votes
0answers
70 views

Spectrum of the Grassmannian Laplacian

The spectrum of the Laplacian (with respect to the Fubini--Study metric) was addressed in this old question. Does anyone know if these results have been extended to the Grassmannians?
5
votes
1answer
205 views

how to prove the $n$-times self-product of a map is null-homotopic

Let $k$ be a fixed positive integer and $\Sigma_k$ the $k$-th symmetric group. By letting $\Sigma_k$ permuting an orthonormal basis of a $k$-dimensional Euclidean space, there is a "regular ...
1
vote
0answers
97 views

Schubert Calculus for the Full Flags

Almost all introductory texts on Schubert calculus discuss the Grassmannian case only. Does there exist a nice discussion of the full flag manifold case $SU(N)/T^{N-1}$? A low dimensional example ...
2
votes
0answers
96 views

obstructions to embeddings of manifolds into Grassmannians

Let $G_k(\mathbb{R}^n)$ be the Grassmannian consisting of $k$-dimensional subspaces in $\mathbb{R}^n$ and $AG_k(\mathbb{R}^n)$ the "affine Grassmannian" consisting of $k$-dimensional planes in ...
2
votes
1answer
187 views

Is this affine-subspace analogue of a Grassmannian a classifying space?

Let $AG_k(\mathbb{R}^N)$ be the "affine Grassmannian" consisting of $k$-dimensional hyperplanes (i.e. affine subspaces) in $\mathbb{R}^N$. Is there any relation between $AG_k(\mathbb{R}^N)$ and the ...
4
votes
3answers
191 views

A natural embedding of the total space of tautological bundle over $G(2,n)$ in $G(2,n+1)$

I learned from the following post that the total space of the tautological line bundle over $\mathbb{R}P^{n}$ is diffeomorphic to $\mathbb{R}P^{n+1}\setminus \{pt\}$.(There is a natural ...
6
votes
1answer
168 views

Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?

Let $$ M:=\{(P,\pi)\mid P\not\in\pi\}\subset\mathbb{P}^n\times\mathbb{P}^{n\,\ast} $$ be the open and dense (and as such $2n$-dimensional) subset of non-incident point-hyperplane pairs. If ...
4
votes
1answer
159 views

Parametrization of Schubert varieties in isotropic Grassmannians by partitions

Let $X=\mathbb{G}_Q(l,p)$ be the isotropic Grassmannian, where $l\leq p-2$. Let $q=p-l$. Let $W^P$ be the set of minimal length representatives. Let $\tilde{\mathcal{Q}}(l,p)$ be the set of partition ...
4
votes
2answers
213 views

Notes on flag varieties and Grassmannians for beginners

Can you suggest books or lecture notes (for beginners) covering basic material about flag varieties and Grassmannians (of reductive groups), with emphasis on the usual flag variety, i.e. flag variety ...
0
votes
0answers
65 views

construction of grassmannian manifolds as collection of subspaces of Euclidean space

The grassmannian $G_k(\mathbb{R}^n)$ is the collection of all $k$-dimensional linear subspaces of $\mathbb{R}^n$ equipped with the quotient topology. The cohomology ring of $G_k(\mathbb{R}^n)$ has ...
2
votes
0answers
49 views

Stiefel manifolds and “simplicial complex chromated Sitefel manifolds”

Let $K$ be a simplicial complex whose vertices are labelled by $1,2,\cdots,k$. I want to define a variant concept of the open Stiefel manifolds $$ ...
2
votes
1answer
202 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
180 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
273 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
111 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
147 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
277 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 ...
18
votes
1answer
356 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
117 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
1answer
166 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 ...
2
votes
1answer
173 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
218 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 ...
1
vote
0answers
96 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
431 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
47 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
112 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
105 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
367 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
127 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
317 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
469 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
207 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
129 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
209 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
431 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
55 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
193 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
141 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
89 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 ...
58
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 ...
8
votes
2answers
431 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 ...
8
votes
2answers
527 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
110 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 ...
2
votes
0answers
415 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
539 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
147 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 $V$ ...